This paper studies the Hamiltonian coloring and Hamiltonian chromatic number for different graphs .the main
results are1.For any integer n greater than or equal to three, Hamiltonian chromatic number of Cn is equal to n-2.
2. G is a graph obtained by adding a pendant edge to Hamiltonian graph H, and then Hamiltonian chromatic
number of G is equal to n-1. 3. For every connected graph G of order n greater than or equal to 2, Hamiltonian
chromatic number of G is not more than one increment of square of (n-2).
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
1) This document discusses graph colouring and related concepts like chromatic number, independent sets, cliques, greedy algorithms, degeneracy, Mycielski's construction, critical graphs, and counting colourings.
2) Key results include Brooks' Theorem which gives an upper bound for chromatic number based on maximum degree, and Gallai-Roy-Vitaver Theorem relating chromatic number to longest directed path in an orientation.
3) Mycielski's construction is introduced as a way to increase the chromatic number of a graph by 1 while preserving being triangle-free.
The Fundamental Solution of an Extension to a Generalized Laplace EquationJohnathan Gray
This thesis examines the fundamental solution to a generalized Laplace equation in a three-dimensional sub-Riemannian space called Grushin space. The author presents the main result as a theorem stating that the generalized Laplace operator applied to a particular function f equals zero, with f containing vector fields and functions that define the Grushin space. The proof of the theorem involves directly computing the generalized Laplace operator by taking derivatives of f with respect to the vector fields. This generalizes previous work that solved the two-dimensional case.
Aieee 2003 maths solved paper by fiitjeeMr_KevinShah
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -i.
This document contains a 50 question multiple choice test on quantitative aptitude topics including ratio and proportion, equations, simple and compound interest, permutations and combinations, sequences and series, limits and calculus, statistics, probability, sampling theory, and index numbers. The questions cover defining key terms, solving equations, evaluating integrals and limits, probability calculations, and data analysis concepts. The answer key is provided at the end.
On sum edge coloring of regular, bipartite and split graphs政謙 陳
This document summarizes research on edge coloring of regular, bipartite, and split graphs. It begins with definitions of proper edge coloring and chromatic index. It then discusses results on edge chromatic sums of regular graphs, including that there is a polynomial time approximation algorithm for finding edge chromatic sums of r-regular graphs for r ≥ 3. The document also discusses edge chromatic sums of bipartite graphs, showing that certain problems related to finding U-sequential colorings and edge chromatic sums are NP-complete for bipartite graphs with maximum degree 3.
This document summarizes research on extremal graphs without three-cycles or four-cycles. The authors derive theoretical upper and lower bounds on the maximum number of edges f(v) in graphs of order v that contain no three-cycles or four-cycles. They provide the exact values of f(v) for all v up to 24 and constructive lower bounds for f(v) up to 200. The document also defines restricted tree structures that are useful in analyzing extremal graphs and establishes properties of these graphs.
This document summarizes research on using elliptic curve cryptography based on imaginary quadratic orders. It shows that for elliptic curves over a finite field Fq, if q satisfies certain conditions, the elliptic curve discrete logarithm problem can be reduced to the discrete logarithm problem over the finite field Fp2. This allows the elliptic curve discrete logarithm problem to potentially be solved faster. It then provides examples of how to construct "weak curves" that satisfy the necessary conditions.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
1) This document discusses graph colouring and related concepts like chromatic number, independent sets, cliques, greedy algorithms, degeneracy, Mycielski's construction, critical graphs, and counting colourings.
2) Key results include Brooks' Theorem which gives an upper bound for chromatic number based on maximum degree, and Gallai-Roy-Vitaver Theorem relating chromatic number to longest directed path in an orientation.
3) Mycielski's construction is introduced as a way to increase the chromatic number of a graph by 1 while preserving being triangle-free.
The Fundamental Solution of an Extension to a Generalized Laplace EquationJohnathan Gray
This thesis examines the fundamental solution to a generalized Laplace equation in a three-dimensional sub-Riemannian space called Grushin space. The author presents the main result as a theorem stating that the generalized Laplace operator applied to a particular function f equals zero, with f containing vector fields and functions that define the Grushin space. The proof of the theorem involves directly computing the generalized Laplace operator by taking derivatives of f with respect to the vector fields. This generalizes previous work that solved the two-dimensional case.
Aieee 2003 maths solved paper by fiitjeeMr_KevinShah
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -i.
This document contains a 50 question multiple choice test on quantitative aptitude topics including ratio and proportion, equations, simple and compound interest, permutations and combinations, sequences and series, limits and calculus, statistics, probability, sampling theory, and index numbers. The questions cover defining key terms, solving equations, evaluating integrals and limits, probability calculations, and data analysis concepts. The answer key is provided at the end.
On sum edge coloring of regular, bipartite and split graphs政謙 陳
This document summarizes research on edge coloring of regular, bipartite, and split graphs. It begins with definitions of proper edge coloring and chromatic index. It then discusses results on edge chromatic sums of regular graphs, including that there is a polynomial time approximation algorithm for finding edge chromatic sums of r-regular graphs for r ≥ 3. The document also discusses edge chromatic sums of bipartite graphs, showing that certain problems related to finding U-sequential colorings and edge chromatic sums are NP-complete for bipartite graphs with maximum degree 3.
This document summarizes research on extremal graphs without three-cycles or four-cycles. The authors derive theoretical upper and lower bounds on the maximum number of edges f(v) in graphs of order v that contain no three-cycles or four-cycles. They provide the exact values of f(v) for all v up to 24 and constructive lower bounds for f(v) up to 200. The document also defines restricted tree structures that are useful in analyzing extremal graphs and establishes properties of these graphs.
This document summarizes research on using elliptic curve cryptography based on imaginary quadratic orders. It shows that for elliptic curves over a finite field Fq, if q satisfies certain conditions, the elliptic curve discrete logarithm problem can be reduced to the discrete logarithm problem over the finite field Fp2. This allows the elliptic curve discrete logarithm problem to potentially be solved faster. It then provides examples of how to construct "weak curves" that satisfy the necessary conditions.
Hosoya polynomial, wiener and hyper wiener indices of some regular graphsieijjournal
Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest
path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G,
whereas the hyper-Wiener index WW(G) is defined as ( ) ( ( ) ( ) ) ( )
2
{u,v} V G
WW G d v,u d v,u .
Î
= + Also, the
Hosoya polynomial was introduced by H. Hosoya and define ( ) ( )
( )
,
{u,v} V G
, . d v u H G x x
Î
= In this
paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are
determined.
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
This document contains a 75 question test paper covering topics in mathematics including algebra, trigonometry, coordinate geometry, and calculus. The test has 90 minutes allotted and covers topics such as functions, quadratic equations, trigonometric identities, binomial coefficients, and geometric concepts like angles, triangles, and coordinate planes.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
(Www.entrance exam.net)-sail placement sample paper 5SAMEER NAIK
This document provides a 75 question multiple choice test paper with questions ranging in topics from algebra, trigonometry, geometry, and calculus. The test is allotted 90 minutes and covers concepts such as functions, equations, properties of circles, ellipses, parabolas, and hyperbolas, trigonometric identities, and geometric shapes. Several questions involve finding lengths, areas, angles between lines, points of intersection, tangents, and properties of conic sections.
Finite fields are important in cryptography. They involve arithmetic operations on a finite set of elements called a Galois field. Two important examples are GF(p), the integers modulo a prime p, and GF(2n), polynomials modulo an irreducible polynomial of degree n with coefficients in GF(2). Polynomial arithmetic in GF(2n) can be performed using bitwise XOR and shift operations, with modulo reduction using the irreducible polynomial. This allows efficient computation in finite fields important for cryptography.
A linear algorithm for the grundy number of a treeijcsit
A coloring of a graph G = (V ,E) is a partition {V1, V2, . . . , Vk} of V into independent sets or color classes.
A vertex v
Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j <i.
A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy
number of a graph is the maximum number of colors in a Grundy coloring. We derive a natural upper
bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound.
In particular, this gives a linear-time algorithm to determine the Grundy number of a tree.
IRJET- On Greatest Common Divisor and its Application for a Geometrical S...IRJET Journal
This document discusses greatest common divisors (GCD) and their application to geometric structures. It begins by defining GCD and least common multiple (LCM) and proving several properties about them. It then derives formulas for the minimum number of identical rectangular tiles or cuboids needed to construct a square floor or cube, respectively. Specifically, it shows the minimum number of tiles/cuboids is equal to the GCD of certain parameters divided by the area/volume of each tile/cuboid. The document provides examples to illustrate the formulas.
This document is the formula booklet for the IB Diploma Programme Mathematical Studies SL course. It contains formulas for various topics in mathematics that students can refer to during their course and examinations. The topics covered include number and algebra, descriptive statistics, logic, sets and probability, geometry and trigonometry, mathematical models, and an introduction to differential calculus. Formulas are provided for calculating percentages, terms of sequences, compound interest, measures of central tendency, truth tables, probability, geometric shapes, trigonometry, and derivatives.
1. The document provides a sample paper for mathematics class 12 with questions covering various topics like vectors, matrices, integrals, probability, linear programming, etc. divided into three sections - short 1 mark questions, 4 mark questions and 6 mark questions.
2. It also includes the marking scheme with answers for all the questions.
3. The questions assess different cognitive levels like remembering, understanding, application and higher order thinking skills.
This document provides a formula booklet for use in the Diploma Programme Mathematics SL course and examinations. It contains formulas for topics covered in the course, including algebra, functions, trigonometry, vectors, statistics, and calculus. The booklet is published by the International Baccalaureate Organization and approved for use from 2014 onwards.
The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.
This document provides an overview of key concepts in vectors and vector algebra covered in an Edexcel FP3 module. It begins with an introduction to vectors and quaternions. Section 1 defines vectors and the vector product. Section 2 defines the triple scalar product and its use in calculating the volume of a tetrahedron. It discusses properties of the vector product, including that it yields a vector perpendicular to the two original vectors and has magnitude equal to the product of the lengths of the original vectors and the sine of the angle between them. It provides examples of using the vector and triple scalar products to calculate the area of a parallelogram and the volume of a tetrahedron.
1. The document is a model question paper with 3 sections containing multiple choice and long answer questions on mathematics.
2. Section A contains 15 multiple choice questions worth 1 mark each. Section B contains 10 long answer questions worth 2 marks each. Section C contains 9 long answer questions worth 5 marks each and 1 compulsory question.
3. The questions cover topics in algebra, trigonometry, geometry, sequences and series, and probability.
List coloring, an intriguing extension of traditional graph coloring, introduces dynamic color assignment by associating each vertex with a list of permissible colors. The list chromatic number, denoted as χl(G), signifies the minimum colors needed to properly color a graph under list constraints. Brooks' Theorem characterizes when a graph is optimally list colorable. Algorithmic techniques, including greedy approaches and backtracking, are pivotal in solving list coloring problems. Applications span various domains such as resource allocation, wireless network frequency assignment, and register allocation. List coloring's link to the classic chromatic number, its variations, computational complexity, and ongoing developments highlight its versatile role in computer science and mathematics, making it a captivating and practical field of study and application.
function f is said to be an even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q and the induced function f* from the edges ofG to {0, 2…….….2(q-1)} defined by f* (uv) = f (u) +f (v) (mod 2q) is bijective. The graph G is said to have an even harmonious labeling. In this paper the even harmonious labeling of a class of graph namely H (2n, 2t+1) is established.
Hosoya polynomial, Wiener and Hyper-Wiener indices of some regular graphsieijjournal
Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈ = + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined. Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈ = + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined.
HOSOYA POLYNOMIAL, WIENER AND HYPERWIENER INDICES OF SOME REGULAR GRAPHSieijjournal
Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈
= + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined.
This document contains solutions to 4 problems from an IMC 2020 Online event. Solution 1 computes the number of words of length n over an alphabet that satisfy certain properties, finding the number is 4n-1+2n-1. Solution 2 uses a recurrence relation approach to derive the same formula. Solution 3 uses generating functions to show the number is also 4n-1+2n-1. Solution 4 shows that if a polynomial p satisfies p(x+1)-p(x)=x100 for all x, then p(1-t)≤p(t) for 0≤t≤1/2.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
The document summarizes some new families of graphs that admit H-cordial and H2-cordial labelings. Specifically:
1) The graphs obtained by duplicating vertices or edges in the cycle Cn for n≥3 admit H2-cordial labelings.
2) The joint sum of two copies of the cycle Cn of even order produces an H2-cordial labeling graph.
3) The split graph of the cycle Cn for even n is H-cordial.
4) The shadow graph D2(Pn) of the path Pn is H-cordial for even n.
The document provides definitions, theorems with proofs, and examples
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
Hosoya polynomial, wiener and hyper wiener indices of some regular graphsieijjournal
Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest
path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G,
whereas the hyper-Wiener index WW(G) is defined as ( ) ( ( ) ( ) ) ( )
2
{u,v} V G
WW G d v,u d v,u .
Î
= + Also, the
Hosoya polynomial was introduced by H. Hosoya and define ( ) ( )
( )
,
{u,v} V G
, . d v u H G x x
Î
= In this
paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are
determined.
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
This document contains a 75 question test paper covering topics in mathematics including algebra, trigonometry, coordinate geometry, and calculus. The test has 90 minutes allotted and covers topics such as functions, quadratic equations, trigonometric identities, binomial coefficients, and geometric concepts like angles, triangles, and coordinate planes.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
(Www.entrance exam.net)-sail placement sample paper 5SAMEER NAIK
This document provides a 75 question multiple choice test paper with questions ranging in topics from algebra, trigonometry, geometry, and calculus. The test is allotted 90 minutes and covers concepts such as functions, equations, properties of circles, ellipses, parabolas, and hyperbolas, trigonometric identities, and geometric shapes. Several questions involve finding lengths, areas, angles between lines, points of intersection, tangents, and properties of conic sections.
Finite fields are important in cryptography. They involve arithmetic operations on a finite set of elements called a Galois field. Two important examples are GF(p), the integers modulo a prime p, and GF(2n), polynomials modulo an irreducible polynomial of degree n with coefficients in GF(2). Polynomial arithmetic in GF(2n) can be performed using bitwise XOR and shift operations, with modulo reduction using the irreducible polynomial. This allows efficient computation in finite fields important for cryptography.
A linear algorithm for the grundy number of a treeijcsit
A coloring of a graph G = (V ,E) is a partition {V1, V2, . . . , Vk} of V into independent sets or color classes.
A vertex v
Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j <i.
A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy
number of a graph is the maximum number of colors in a Grundy coloring. We derive a natural upper
bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound.
In particular, this gives a linear-time algorithm to determine the Grundy number of a tree.
IRJET- On Greatest Common Divisor and its Application for a Geometrical S...IRJET Journal
This document discusses greatest common divisors (GCD) and their application to geometric structures. It begins by defining GCD and least common multiple (LCM) and proving several properties about them. It then derives formulas for the minimum number of identical rectangular tiles or cuboids needed to construct a square floor or cube, respectively. Specifically, it shows the minimum number of tiles/cuboids is equal to the GCD of certain parameters divided by the area/volume of each tile/cuboid. The document provides examples to illustrate the formulas.
This document is the formula booklet for the IB Diploma Programme Mathematical Studies SL course. It contains formulas for various topics in mathematics that students can refer to during their course and examinations. The topics covered include number and algebra, descriptive statistics, logic, sets and probability, geometry and trigonometry, mathematical models, and an introduction to differential calculus. Formulas are provided for calculating percentages, terms of sequences, compound interest, measures of central tendency, truth tables, probability, geometric shapes, trigonometry, and derivatives.
1. The document provides a sample paper for mathematics class 12 with questions covering various topics like vectors, matrices, integrals, probability, linear programming, etc. divided into three sections - short 1 mark questions, 4 mark questions and 6 mark questions.
2. It also includes the marking scheme with answers for all the questions.
3. The questions assess different cognitive levels like remembering, understanding, application and higher order thinking skills.
This document provides a formula booklet for use in the Diploma Programme Mathematics SL course and examinations. It contains formulas for topics covered in the course, including algebra, functions, trigonometry, vectors, statistics, and calculus. The booklet is published by the International Baccalaureate Organization and approved for use from 2014 onwards.
The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.
This document provides an overview of key concepts in vectors and vector algebra covered in an Edexcel FP3 module. It begins with an introduction to vectors and quaternions. Section 1 defines vectors and the vector product. Section 2 defines the triple scalar product and its use in calculating the volume of a tetrahedron. It discusses properties of the vector product, including that it yields a vector perpendicular to the two original vectors and has magnitude equal to the product of the lengths of the original vectors and the sine of the angle between them. It provides examples of using the vector and triple scalar products to calculate the area of a parallelogram and the volume of a tetrahedron.
1. The document is a model question paper with 3 sections containing multiple choice and long answer questions on mathematics.
2. Section A contains 15 multiple choice questions worth 1 mark each. Section B contains 10 long answer questions worth 2 marks each. Section C contains 9 long answer questions worth 5 marks each and 1 compulsory question.
3. The questions cover topics in algebra, trigonometry, geometry, sequences and series, and probability.
List coloring, an intriguing extension of traditional graph coloring, introduces dynamic color assignment by associating each vertex with a list of permissible colors. The list chromatic number, denoted as χl(G), signifies the minimum colors needed to properly color a graph under list constraints. Brooks' Theorem characterizes when a graph is optimally list colorable. Algorithmic techniques, including greedy approaches and backtracking, are pivotal in solving list coloring problems. Applications span various domains such as resource allocation, wireless network frequency assignment, and register allocation. List coloring's link to the classic chromatic number, its variations, computational complexity, and ongoing developments highlight its versatile role in computer science and mathematics, making it a captivating and practical field of study and application.
function f is said to be an even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q and the induced function f* from the edges ofG to {0, 2…….….2(q-1)} defined by f* (uv) = f (u) +f (v) (mod 2q) is bijective. The graph G is said to have an even harmonious labeling. In this paper the even harmonious labeling of a class of graph namely H (2n, 2t+1) is established.
Hosoya polynomial, Wiener and Hyper-Wiener indices of some regular graphsieijjournal
Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈ = + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined. Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈ = + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined.
HOSOYA POLYNOMIAL, WIENER AND HYPERWIENER INDICES OF SOME REGULAR GRAPHSieijjournal
Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈
= + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined.
This document contains solutions to 4 problems from an IMC 2020 Online event. Solution 1 computes the number of words of length n over an alphabet that satisfy certain properties, finding the number is 4n-1+2n-1. Solution 2 uses a recurrence relation approach to derive the same formula. Solution 3 uses generating functions to show the number is also 4n-1+2n-1. Solution 4 shows that if a polynomial p satisfies p(x+1)-p(x)=x100 for all x, then p(1-t)≤p(t) for 0≤t≤1/2.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
The document summarizes some new families of graphs that admit H-cordial and H2-cordial labelings. Specifically:
1) The graphs obtained by duplicating vertices or edges in the cycle Cn for n≥3 admit H2-cordial labelings.
2) The joint sum of two copies of the cycle Cn of even order produces an H2-cordial labeling graph.
3) The split graph of the cycle Cn for even n is H-cordial.
4) The shadow graph D2(Pn) of the path Pn is H-cordial for even n.
The document provides definitions, theorems with proofs, and examples
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
1. The document provides various formulas related to mathematics including algebra, trigonometry, geometry, calculus, probability, and statistics.
2. Formulas are given for topics like ratios, proportions, arithmetic and geometric progressions, simple and compound interest, mixtures and allegations, quadratic equations, triangles, circles, solids and their surface areas and volumes.
3. Permutations, combinations, and formulas from number systems, modern algebra and basic linear algebra are also summarized.
The document contains the solutions to 5 problems from the 2017 Canadian Mathematical Olympiad. The first problem involves using an inequality to prove that the sum of fractions involving three non-negative real numbers is greater than 2. The second problem relates the number of divisors of a positive integer to a function and proves if the input is prime, the output is also prime. The third problem counts the number of balanced subsets of numbers and proves the count is odd.
Applications and Properties of Unique Coloring of GraphsIJERA Editor
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
The document discusses graph theory and provides definitions and examples of various graph concepts. It defines what a graph is consisting of vertices and edges. It also defines different types of graphs such as simple graphs, multigraphs, digraphs and provides examples. It discusses graph terminology, models, degree of graphs, handshaking lemma, special graphs and applications. It also provides explanations of planar graphs, Euler's formula and graph coloring.
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -1.
From planar maps to spatial topology change in 2d gravityTimothy Budd
The document summarizes a talk on generalized causal dynamical triangulations (CDT) in two dimensions. It introduces the generalized CDT model, which allows spatial topology to change in time. It describes how generalized CDT can be solved by viewing causal quadrangulations as labeled trees. It also discusses bijections between labeled quadrangulations and labeled planar maps via Schaeffer's algorithm, which allow counting the numbers of objects in the generalized CDT model.
This document summarizes Dorian Ehrlich's thesis investigating the quantum case of Horn's question. Horn's question asks for which sequences of real numbers γ can there exist Hermitian matrices A, B, and C=A+B with eigenvalues α, β, and γ. Knutson and Tao provided a solution using intersections of Schubert varieties. Ehrlich seeks to generalize this to intersections linked by smooth curves, known as the quantum case. The document provides background on Hermitian matrices, Schubert varieties, and Knutson and Tao's solution. It then outlines Ehrlich's plan to prove a "quantum analogue" of Knutson and Tao's saturation conjecture to address the quantum case of Horn's question.
The document summarizes research on finding polychromatic solutions to linear equations in an r-bounded coloring of the natural numbers. The research builds on previous work exploring rainbow analogues of classical Ramsey theory results. Key findings include:
1) Theorems were proven showing that for equations of the form ax-by=c with integers a,b,c and gcd(a,b)|c, there exists a polychromatic solution in any r-bounded coloring.
2) Explicit recurrence relations and formulas were defined to generate chains of solutions to equations with two variables.
3) The results for two variables were used to show polychromatic solutions exist for equations with three variables of the form a1
The document summarizes research on the parameterized complexity of the graph MOTIF problem. It begins by defining the problem and providing an example. It then discusses how graph MOTIF can be solved efficiently using different parameters, such as cluster editing, distance to clique, and vertex cover number. The document also analyzes parameters for which graph MOTIF remains NP-hard, such as the deletion set number parameter. In conclusion, it provides references for the algorithms and results discussed.
This document defines k-rankings of graphs and 2-rankings specifically. It summarizes findings on the 2-ranking numbers of various graphs, including hypercubes (χ2(Qn) = n + 1), Cartesian products of complete graphs (χ2(KmKn) ≥ nHm), and subcubic graphs (χ2(G) ≤ 7). It also presents a probabilistic construction showing that for some graphs G with maximum degree Δ(G) ≤ k, the 2-ranking number is Ω(k2/logk).
Similar to Hamiltonian Chromatic Number of Graphs (20)
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...PriyankaKilaniya
Energy efficiency has been important since the latter part of the last century. The main object of this survey is to determine the energy efficiency knowledge among consumers. Two separate districts in Bangladesh are selected to conduct the survey on households and showrooms about the energy and seller also. The survey uses the data to find some regression equations from which it is easy to predict energy efficiency knowledge. The data is analyzed and calculated based on five important criteria. The initial target was to find some factors that help predict a person's energy efficiency knowledge. From the survey, it is found that the energy efficiency awareness among the people of our country is very low. Relationships between household energy use behaviors are estimated using a unique dataset of about 40 households and 20 showrooms in Bangladesh's Chapainawabganj and Bagerhat districts. Knowledge of energy consumption and energy efficiency technology options is found to be associated with household use of energy conservation practices. Household characteristics also influence household energy use behavior. Younger household cohorts are more likely to adopt energy-efficient technologies and energy conservation practices and place primary importance on energy saving for environmental reasons. Education also influences attitudes toward energy conservation in Bangladesh. Low-education households indicate they primarily save electricity for the environment while high-education households indicate they are motivated by environmental concerns.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
This study Examines the Effectiveness of Talent Procurement through the Imple...DharmaBanothu
In the world with high technology and fast
forward mindset recruiters are walking/showing interest
towards E-Recruitment. Present most of the HRs of
many companies are choosing E-Recruitment as the best
choice for recruitment. E-Recruitment is being done
through many online platforms like Linkedin, Naukri,
Instagram , Facebook etc. Now with high technology E-
Recruitment has gone through next level by using
Artificial Intelligence too.
Key Words : Talent Management, Talent Acquisition , E-
Recruitment , Artificial Intelligence Introduction
Effectiveness of Talent Acquisition through E-
Recruitment in this topic we will discuss about 4important
and interlinked topics which are
Open Channel Flow: fluid flow with a free surfaceIndrajeet sahu
Open Channel Flow: This topic focuses on fluid flow with a free surface, such as in rivers, canals, and drainage ditches. Key concepts include the classification of flow types (steady vs. unsteady, uniform vs. non-uniform), hydraulic radius, flow resistance, Manning's equation, critical flow conditions, and energy and momentum principles. It also covers flow measurement techniques, gradually varied flow analysis, and the design of open channels. Understanding these principles is vital for effective water resource management and engineering applications.
Blood finder application project report (1).pdfKamal Acharya
Blood Finder is an emergency time app where a user can search for the blood banks as
well as the registered blood donors around Mumbai. This application also provide an
opportunity for the user of this application to become a registered donor for this user have
to enroll for the donor request from the application itself. If the admin wish to make user
a registered donor, with some of the formalities with the organization it can be done.
Specialization of this application is that the user will not have to register on sign-in for
searching the blood banks and blood donors it can be just done by installing the
application to the mobile.
The purpose of making this application is to save the user’s time for searching blood of
needed blood group during the time of the emergency.
This is an android application developed in Java and XML with the connectivity of
SQLite database. This application will provide most of basic functionality required for an
emergency time application. All the details of Blood banks and Blood donors are stored
in the database i.e. SQLite.
This application allowed the user to get all the information regarding blood banks and
blood donors such as Name, Number, Address, Blood Group, rather than searching it on
the different websites and wasting the precious time. This application is effective and
user friendly.
OOPS_Lab_Manual - programs using C++ programming language
Hamiltonian Chromatic Number of Graphs
1. Dr. B. Ramireddy et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 1) January 2016, pp.33-35
www.ijera.com 33|P a g e
Hamiltonian Chromatic Number of Graphs
Dr. B. Ramireddy1
, U. Mohan Chand2
, A.Sri Krishna Chaitanya3
, Dr. B.R.
Srinivas4
1) Professor & H.O.D, Hindu College, Guntur, (A.P.) INDIA.
2) Associate Professor of Mathematics & H.O.D, Rise Krishna Sai Prakasam Group of Institutions, Ongole,
(A.P) INDIA.
3) Associate Professor of Mathematics & H.O.D, Chebrolu Engineering College, Chebrolu, Guntur Dist. (A.P)
4) Professor of Mathematics, St. Mary’s Group of Institutions, Chebrolu, Guntur Dist. (A.P) INDIA.
Abstract
This paper studies the Hamiltonian coloring and Hamiltonian chromatic number for different graphs .the main
results are1.For any integer n greater than or equal to three, Hamiltonian chromatic number of Cn is equal to n-2.
2. G is a graph obtained by adding a pendant edge to Hamiltonian graph H, and then Hamiltonian chromatic
number of G is equal to n-1. 3. For every connected graph G of order n greater than or equal to 2, Hamiltonian
chromatic number of G is not more than one increment of square of (n-2).
Mathematics Subject Classification 2000: 03Exx, 03E10, 05CXX, 05C15, 05C45.
Key words: Chromatic number, Hamiltonian coloring, Hamiltonian chromatic number, pendant edge,
spanning connected graph.
I. Introduction
Generally in a (d – 1) radio coloring of a
connected graph G of diameter d, the colors assigned
to adjacent vertices must differ by at least d-1, the
colors assigned to two vertices whose distance is 2
must differ by at least d-2, and so on up to antipodal
vertices, whose colors are permitted to be the same.
For this reason, (d-1) radio colorings are also referred
to as antipodal colorings.
In the case of an antipodal coloring of the path Pn
of order n > 2, only the two end-vertices are
permitted to be colored the same. If u and v are
distinct vertices of Pn and d (u, v) = i, then |c (u)-c
(v)| > n – 1 – i. Since Pn is a tree, not only is i the
length of a shortest u – u path in Pn, it is the length of
the only u – v path in Pn. In particular, is the length of
a longest u – v path?
The detour distance D (u, v) between two
vertices u and v in a connected graph G is defined as
the length of a longest u – v path in G. Hence the
length of a longest u – v path in Pn is D (u, v) = d (u,
v). Therefore, in the case of the path Pn, an antipodal
coloring of Pn can also be defined as a vertex coloring
c that satisfies.
D (u,v) + |c(v)| > n – 1, for every two distinct
vertices u and v of Pn.
§1.1 Definition: Vertex coloring c that satisfy were
extended from paths of order n to arbitrary connected
graphs of order n by Gary Chartrand, Ladislav
Nebesky, and Ping Zhang .A Hamiltonian coloring
of a connected graph G of order n is a vertex coloring
c such that, D (u,v) + |c(v)| > n – 1, for every tow
distinct vertices u and v of G. the largest color
assigned to a vertex of G by c is called the value of c
and is denoted by hc(c). The Hamiltonian
chromatic number hc (G) is the smallest value
among all Hamiltonian colorings of G.
EX: Figure.1 (a) shows a graph H of order 5. A
vertex coloring c of H is shown in Figure.1 (b). Since
D (u, v) + |c (u) - c (v)| > 4 for every two distinct
vertices u and u of H, it follows that c is a
Hamiltonian coloring and so hc(c) =4. Hence hc (H)
> 4. Because no two of the vertices t, w, x, and y are
connected by a Hamiltonian path, these must be
assigned distinct colors and so hc (H) > 4. Thus hc
(H) = 4.
1. A graph with Hamiltonian chromatic number 4
If a connected graph G of order n has
Hamiltonian chromatic number 1, then D(u,v) = n – 1
for every two distinct vertices u and v of G and
consequently G is Hamiltonian-connected, that is,
every two vertices of G are connected by a
RESEARCH ARTICLE OPEN ACCESS
2. Dr. B. Ramireddy et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 1) January 2016, pp.33-35
www.ijera.com 34|P a g e
Hamiltonian pat. Indeed, hc (G) = 1 if and only if G
is Hamiltonian – connected. Therefore, the
Hamiltonian Chromatic Number of a connected
graph G can be considered as a measure of how close
G is to being Hamiltonian connected. That is the
closer hc (G) is to 1, the closer G is to being
Hamiltonian connected. The three graphs H1, H2 and
H3 shown in below figure 2 are all close (in this
sense) to being Hamiltonian-connected since hc (Hi)
= 2 for i = 1, 2, 3.
H1 H2
H3
2. Three graphs with Hamiltonian chromatic number
2
II. Theorem: For every integer n > 3, hc
(K1,n-1) = (n-2)2
+ 1
Proof: Since hc (K 1, 2) = 2 (See H1 in Figure 2) we
may assume that n ≥ 4.
Let G = K 1,n-1 where V(G) = {v1,v2…..vn} and vn is
the central vertex. Define the coloring c of G by c
(vn) = 1 and C (vi ) = (n-1) + (i-1)(n-3) for 1 ≤ i ≤ n -
1.Then c is a Hamiltonian coloring of G and
hc (G) ≤ hc (c) = c (vn-1) = (n-1) + (n-2) (n-3) = (n-2)2
+ 1.It remains to show that hc (G) ≥ (n-2)2
+ 1.
Let c be a Hamiltonian coloring of G such that hc(c)
= hc (G). Because G contains no Hamiltonian path, c
assigns distinct colors to the vertices of G. We may
assume that C (v1) < c (v2) < … < c (vn-1). We now
consider three cases, depending on the color assigned
to the central vertex vn.
Case 1.
c (vn) = 1.
Since
D (v1, vn) = 1 and D (vi, vi+1) = 2 for 1 ≤ i ≤ n-2.
It follows that
C (vn-1) ≥ 1 + (n-2) + (n-2) (n-3) = (n-2)2
+ 1
And so
hc (G) = hc(c) = c (vn-1) ≥ (n-2)2
+ 1.
Case 2.
C (vn) = hc(c)
Thus, in this case,
1 = c (v1) < c (v2) < … < c (vn-1) < c (vn)
Hence
C (vn) ≥ 1 + (n-2)(n-3) + (n-2) = (n-2)2
+ 1
And so
hc (G) = hc (v) = c(vn) ≥ (n-2)2
+1.
Case 3.
C (vj) < c (vn) < c (vj+1) for some integer j with 1 ≤ j ≤
n – 2.
Thus c(v1) = 1 and c(vn-1) = hc(c).
in this case
C (Vj) ≥ 1 + (j-1) (n-3),
C (vn) ≥ c (vj) + (n-2)
C (vj+1) ≥ c (vn) + (n-2) and
C (vn-1) ≥ c (vj+1) + [(n-1) – (j+1)] (n-3).
Therefore,
C (vn-1) ≥ 1 + (j-1)(n-3) + 2(n-2) + (n-j-2)(n-3)
= (2n-3) + (n-3)2
= (n-2)2
+ 2 > (n-2)2
+ 1.
And so hc (G) = hc(c) = c (vn-1) > (n-2)2 +
1.
Hence in any case,
hc (G) ≥ (n-2)2
+ 1 and so hc (G) = (n-2)2
+ 1.
III. Theorem: For every integer n > 3, hc
(Cn) = n-2.
Proof.
Since we noted that hc (Cn) = n-2 for n = 3, 4, 5.
We may assume that n ≥ 6. Let Cn = (v1, v2…vn , v1).
Because the vertex coloring c of Cn defined by
c (v1) = c(v2) = 1, c(vn-1) = c(vn) = n-2 and c(vi) = i-1
for 3≤i≤n-2 is a Hamiltonian coloring, it follows that
hc(Cn) ≤ n-2.Assume, to the contrary , that hc (Cn) <
n-2 for some integer n ≥ 6. Then there exists a
Hamiltonian (n-3) coloring c of Cn. We consider two
cases, according to whether n is odd or n is even.
Case 1.
n is odd: Then n = 2k + 1 for some integer k ≥ 3.
Hence there exists a Hamiltonian (2k-2) coloring c of
Cn. Let,
A = {1, 2… k-1} and B = {k, k+1…2k-2}
For every vertex u of Cn, there are two vertices v of
Cn such that D (u,v) is minimum (and d(u,v) is
maximum), namely D(u,v) = d(u,v) + 1 = k+1. For u
= vi, these two vertices v are vi+k and vi+k+1 (where the
addition in i + k and i + k + 1 is performed modulo
n).Since c is a Hamiltonian coloring.
D (u, v) + |c (u) – c (v)| > n =l =2k.BecauseD (u, v) =
k + 1, it follows that
|c(u) – c(v)| > k – 1.
Therefore, if c (u) A, then the colors of these two
vertices v with this property must belong to B. In
particular, if c (vi) A, then (vi+k) B. Suppose that
there are a vertices of Cn whose colors belong to A
and b vertices of Cn whose colors belong to B. Then
b>a However, if c (vi) B, then c (vi+k) belongs to a
implying that a > b and so. a=b. since a + b = n
and n is odd, this is impossible.
Case 2.
n is even: Then n = 2k for some integer k > 3. Hence
there exists a Hamiltonian (2k-3) - coloring c of Cn.
For every vertex u of Cn, there is a unique vertex v of
Cn for which D (u, v) is minimum (and d (u, v) is
maximum), namely, d (u, v) = k. For u = vi, this
3. Dr. B. Ramireddy et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 1) January 2016, pp.33-35
www.ijera.com 35|P a g e
vertex v is vi+k (where the addition in i + k is
performed modulo n).
Since c is a Hamiltonian coloring, D (u,v) + |c(u) –
c(v)| > n – 1 = 2k – 1. Because D (u, v) = k, it follows
that |c (u) – c (v)| > k – 1. This implies, however, that
if
c (u) = k-1, then there is no color that can be assigned
to u to satisfy this requirement. Hence no vertex of Cn
can be assigned the color k- 1 by c.
Let, A = {1, 2… k-2} and B = {k, k+1…2k-3}.
Thus |A| = |B| = k – 2. If c (vi) A, then c
(vi+k) B. Also, if c (vi) B, then c (vi+k) A. Hence
there are k vertices of Cn assigned colors from
B.Consider two adjacent vertices of Cn, one of which
is assigned a color from A and the other is assigned a
color from B. We may assume that c (v1) A and c
(v2) B. Then c (vk+1) B. Since D (v2,vk+1) = k+1,
it follows that |c (v2) – c(vk+1) > k – 2. Because c (v2),
c (vk+1) B, this implies that one of c (v2) and c
(vk+1) is at least 2k-2. This is a contradiction.
§ 3.1Proposition: If H is a spanning connected sub
graph of a graph G, then hc (G) < hc (H)
Proof.
Suppose that the order of H is n. Let c be a
Hamiltonian coloring of H such that hc(c) hc (H).
Then DH (u,v) + |c(u) – c(v)| > n -1 for every two
distinct vertices u and v of H. since DG (u,v) > DH
(u,v) for every two distinct vertices u and v of H, it
follows that DG (u,v) + |c(u) – c(v)| > n – 1 and so c
is a Hamiltonian coloring of G as well. Hence hc (G)
< hc (c) = hc (H).
§ 3.2 Proposition: Let H be a Hamiltonian graph
of order n – 1 > 3. If G is a graph obtained by
adding a pendant edge to H, then he (G) = n – 1.
Proof. Suppose that C = (v1,v2…vn-1,v1) is a
Hamiltonian cycle of H and v1vn is the pendant edge
of G. Let c be a Hamiltonian coloring of G. Since DG
(u, v) < n-2 for every two distinct vertices u and v of
C, no two vertices of C can be assigned the same
color by c. Consequently, hc (c) > n – 1 and so hc (G)
> n – 1.
Now define a coloring c` of G by
C1
(vi) =
We claim that c` is a Hamiltonian coloring of G. First
let vj and vk be two vertices of C where 1 < j < k < n
– 1. The |c1
+ (vj) – c1
(vk)| = k – j and
D (vj,vk) = max {k-j, (n-1) – (k-j)}.
In either case, D (vj,vk) ≥ n-1 + j – k and so
D (vj,vk) + |c1
(vj) – c1
(vk) | ≥ n-1.
For 1≤ j ≤n-1, |c1
(vj) – c1
)vn)| = n-1-j, while
D (vj,vn) ≥ max {j, n-j+1}
And so, D (vj, vn) ≥ j.
Therefore,
D (vj, vn) + |c1
(vj) – c1
(vn)| ≥ n-1.
Hence, as claimed, c’ is a Hamiltonian coloring of G
and so hc (G) ≤ hc (c’) =c1
(vn) = n-1.
IV. Theorem: for every connected graph
G of order n ≥ 2, hc (G) ≤ (n-2)2
+ 1.
Proof. First, if G contains a vertex of degree n-1,
then G contains the star K1,n-1 as a spanning sub
graph. Since hc (K1,n-1) = (n-2)2
+ 1 it follows by
proposition 1 that hc(G) ≤ (n-2)2
+ 1. Hence we may
assume that G contains a spanning tree T that is not a
star and so its complement T contains a Hamiltonian
path P = (v1,v2….vn). Thus vi vi+1 E (T) for 1 ≤ i≤
n-1 and so DT (vi,vi+I) ≥2. Define a vertex coloring c
of T by
C (vi) = (n-2) + (i-2) (n-3) for 1 ≤ i≤ n.
Hence
hc (c) = c (vn) = (n-2) + (n-2) (n-3) = (n-2)2
Therefore, for integers i and j with 1 ≤ i< j ≤ n,
|c (vi) – c (vj)| = (j-i) (n-3).
If j = i+ 1, then
D (vi,vj) + (c (vi) – c (vj)| ≥ 1 + 2(n-3) = 2n-5 ≥ n-1.
Thus c is a Hamiltonian coloring of T. therefore,
hc (G) ≤ hc (T) ≤ hc(c) = c (vn) = (n-2)2
< (n-2)2
+ 1,
Which completes the proof
REFERENCES;
[1] G.Chartrand, L.Nebesky, and P.Zhang,
Hamiltonian graphs. Discrete Appl.Math.
146(2005) 257-272.
[2] O.Ore Note on Hamalton
circuits.Amer.Math.Monthly 67(1960)
[3] P.G Tait, Remarks on the colorings of maps
proc.Royal soc.Edinburgh 10 (1880)
[4] H.Whitney, the Colorings of
Graphs.Ann.Math.33 (1932) 688-718.
[5] D.R.Woodall, List colourings of graphs.
Survey in combinatories 2001 Cambridge
univ. press, Cambridge,(2001) 269-301.
i if 1 < i < n -1
n – 1 if i = n.