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.
ODD GRACEFULL LABELING FOR THE SUBDIVISON OF DOUBLE TRIANGLES GRAPHSijscmcj
The aim of this paper is to present some odd graceful graphs. In particular we show that an odd graceful labeling of the all subdivision of double triangular snakes ( 2∆k -snake ). We also prove that the all subdivision of 2 m∆1-snake are odd graceful. Finally, we generalize the above two results (the all subdivision of 2 m∆k -snake are odd graceful).
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
In [8] Liang and Bai have shown that the - 4 kC snake graph is an odd harmonious graph for each k ³ 1.
In this paper we generalize this result on cycles by showing that the - n kC snake with string 1,1,…,1 when
n º 0 (mod 4) are odd harmonious graph. Also we show that the - 4 kC snake with m-pendant edges for
each k,m ³ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 k mD -
snake are odd harmonious for each k,m ³ 1 . Finally we present some examples to illustrate the proposed
theories.
International Journal of Engineering and Science Invention (IJESI) inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
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.
ODD GRACEFULL LABELING FOR THE SUBDIVISON OF DOUBLE TRIANGLES GRAPHSijscmcj
The aim of this paper is to present some odd graceful graphs. In particular we show that an odd graceful labeling of the all subdivision of double triangular snakes ( 2∆k -snake ). We also prove that the all subdivision of 2 m∆1-snake are odd graceful. Finally, we generalize the above two results (the all subdivision of 2 m∆k -snake are odd graceful).
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
In [8] Liang and Bai have shown that the - 4 kC snake graph is an odd harmonious graph for each k ³ 1.
In this paper we generalize this result on cycles by showing that the - n kC snake with string 1,1,…,1 when
n º 0 (mod 4) are odd harmonious graph. Also we show that the - 4 kC snake with m-pendant edges for
each k,m ³ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 k mD -
snake are odd harmonious for each k,m ³ 1 . Finally we present some examples to illustrate the proposed
theories.
International Journal of Engineering and Science Invention (IJESI) inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
EDGE-NEIGHBOR RUPTURE DEGREE ON GRAPH OPERATIONSmathsjournal
Vulnerability and reliability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links in a communication network. An edge subversion strategy of a graph , say , is a set of edge(s) in whose adjacent vertices which is incident with the removal edge(s) are removed from . The survival subgraph is denoted by . In this paper we give some results for the edge-neighbor-rupture degree of the graph operations and Thorny graph types are examined.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
A graph with n vertex and m edges is said to be cubic graceful labeling if its vertices are labeled with distinct integers {0,1,2,3,……..,m3} such that for each edge f*( uv) induces edge mappings are {13,23,33,……,m3}. A graph admits a cubic graceful labeling is called a cubic graceful graph. In this paper, we proved that < K1,a , K1,b , K1c , K1,d>, associate urp with u(r+1)1 of K1,P, fixed vertices vr and ur of two copies of Pn are cubic graceful.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
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 homogeneous biquadratic diophantineequation x4 y4=17(z2-w2)r2eSAT Journals
Abstract
Five different methods of the non-zero non-negative solutions of non- homogeneous cubic Diophantine equation x4 – y4 = 17( z2 –
w2) R2 are obtained. Some interesting relations among the special numbers and the solutions are exposed.
Keywords: The Method of Factorization, Integer Solutions, Linear Transformation, Relations and Special Numbers
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
EDGE-NEIGHBOR RUPTURE DEGREE ON GRAPH OPERATIONSmathsjournal
Vulnerability and reliability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links in a communication network. An edge subversion strategy of a graph , say , is a set of edge(s) in whose adjacent vertices which is incident with the removal edge(s) are removed from . The survival subgraph is denoted by . In this paper we give some results for the edge-neighbor-rupture degree of the graph operations and Thorny graph types are examined.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
A graph with n vertex and m edges is said to be cubic graceful labeling if its vertices are labeled with distinct integers {0,1,2,3,……..,m3} such that for each edge f*( uv) induces edge mappings are {13,23,33,……,m3}. A graph admits a cubic graceful labeling is called a cubic graceful graph. In this paper, we proved that < K1,a , K1,b , K1c , K1,d>, associate urp with u(r+1)1 of K1,P, fixed vertices vr and ur of two copies of Pn are cubic graceful.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
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 homogeneous biquadratic diophantineequation x4 y4=17(z2-w2)r2eSAT Journals
Abstract
Five different methods of the non-zero non-negative solutions of non- homogeneous cubic Diophantine equation x4 – y4 = 17( z2 –
w2) R2 are obtained. Some interesting relations among the special numbers and the solutions are exposed.
Keywords: The Method of Factorization, Integer Solutions, Linear Transformation, Relations and Special Numbers
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
In [8] Liang and Bai have shown that the kC4 − snake graph is an odd harmonious graph for each k ≥ 1. In this paper we generalize this result on cycles by showing that the kCn − snake with string 1,1,…,1 when n ≡ 0 (mod 4) are odd harmonious graph. Also we show that the kC4 − snake with m-pendant edges for each k,m ≥ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 m∆k - snake are odd harmonious for each k,m ≥ 1 . Finally we present some examples to illustrate the proposed theories.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
New Classes of Odd Graceful Graphs - M. E. Abdel-AalGiselleginaGloria
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ≥ 2, n ≥ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
The i(G)-graph is defined as a graph whose vertex set correspond 1 to 1 with the i(G)-sets of G . Two i(G)- sets say
S1 and 2 S are adjacent in i(G) if there exists a vertex S1
v , and a vertex wS2 such that v is adjacent to w and = { } { } 1 2 S S w v or equivalently = { } { } S2 S1 v w . In this paper we obtain i(G)-graph of some special graphs.
Radix-3 Algorithm for Realization of Type-II Discrete Sine TransformIJERA Editor
In this paper, radix-3 algorithm for computation of type-II discrete sine transform (DST-II) of length N =
3푚 (푚 = 1,2, … . ) is presented. The DST-II of length N can be realized from three DST-II sequences, each of
length N/3. A block diagram of for computation of the radix-3 DST-II algorithm is given. Signal flow graph for
DST-II of length 푁 = 32 is shown to clarify the proposed algorithm.
Radix-3 Algorithm for Realization of Type-II Discrete Sine TransformIJERA Editor
In this paper, radix-3 algorithm for computation of type-II discrete sine transform (DST-II) of length N =
3𝑚 (𝑚 = 1,2, … . ) is presented. The DST-II of length N can be realized from three DST-II sequences, each of
length N/3. A block diagram of for computation of the radix-3 DST-II algorithm is given. Signal flow graph for
DST-II of length 𝑁 = 32 is shown to clarify the proposed algorithm.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
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Knowledge engineering: from people to machines and back
On the 1-2-3-edge weighting and Vertex coloring of complete graph
1. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013
DOI:10.5121/ijcsa.2013.3302 19
ON THE 1-2-3-EDGE WEIGHTING AND
VERTEX COLORING OF COMPLETE GRAPH
Mohammad Reza Farahani1
1
Department of Applied Mathematics, Iran University of Science and Technology (IUST)
Narmak, Tehran, Iran
Mr_Farahani@Mathdep.iust.ac.ir
ABSTRACT
A WEIGHTING OF THE EDGES OF A GRAPH IS CALLED VERTEX-COLORING IF THE LABELED DEGREES OF THE
VERTICES YIELD A PROPER COLORING OF THE GRAPH. IN OTHER WORDS, FOR SOME ,k ∈ LET
F:E(G)→{1,2,…,K} BE AN INTEGER WEIGHTING OF THE EDGES OF A GRAPH G=(V(G);E(G)) WHICH HAVE N
VERTICES AND IMPLIES THAT A VERTEX-COLORING
( )
: ( )v v V G
S f vu∈
= ∑ FOR EVERY VERTEX V∈V(G). IN THIS
PAPER WE OBTAIN FOR K=3 A PROPER 1-2-3-EDGE WEIGHTING AND VERTEX COLORING A FAMILY OF COMPLETE
GRAPHS.
KEYWORDS
Network Protocols, Edge-labeling; Vertex-coloring; Complete Graph.
1. INTRODUCTION
For a graph G=(V(G);E(G)), there exist a function f:E(G)→{1,2,…,k} be an edge weighting of the
edges of G. In other words, for any two arbitrary vertices u,v∈V(G) and edge uv∈E(G) we will
have f(uv)∈{1,2,…,k}. Also, ( )
: ( )v u V G
S f vu∈
= ∑ is a color for a vertex v∈V(G) such that Su≠Sv for
any two arbitrary distinct vertices u,v of G (consider S={Sv}v
∈V(G)) and therefore, a function
s:V(G)→S is a proper vertex-coloring for G.
In 2002, Karonski, Łuczak and Thomason conjectured that such a weighting with k=3 is possible
for all such graphs (see Conjecture 1 and references [8,10]). For k=2 is not sufficient as seen for
instance in complete graphs and cycles of length not divisible by 4. A first constant bound of
k=30 was proved by Addario-Berry, et.al in 2007 [1], which was later improved to K=16 by
Addario-Berry's group in [2] and k=13 by T. Wang and Q. Yu in 2008, [13]. Recently, its new
bounds are k=5 and k=6 by Kalkowski, et.al [8, 9].
In this note we show that there is a proper1-2-3-edge weighting and vertex coloring for a family
K3q for all integer number q of complete graphs and obtain two above functions f and s for K3q
,q∀ ∈ exactly. Thus, we have following theorem that is the main result of this paper.
Theorem 1.1. Consider complete graph K3q with the vertex set V(K3q)={v1,v2,…,v3q} and
the edge set E(K3q)={eij=vivj|vi,vj∈V(K3q)} for every integer number q. Thus there are a
edge weighting f :E(K3q)→ {1,2,3} a vertex coloring s:V(K3q)→S={9q-3,9q-4,…,7q-2,7q-
2. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013
20
3,7q-6,7q-10,…,3q+10,3q+6,3q+3}, such that the induced vertex weights
Si:= 1,
( )j j
n
j i if v v= ≠∑ ∈S.
2. MAIN RESULTS
At first, before prove Theorem 1.1, we contribute the following definition, which is useful to
proving.
Definition 2.1. Let f be a function that obtained from k-edge weighting and vertex coloring of a
connected graph G as order n. By using f, we can part the edge set E(G) into k important sets
E(G)i, k∀ ∈ and are equal to E(G)i={vivj
∈E(G)|f(vivj)=i} ⇒|E(G)i|=γi. Therefore m=|E(G)|=Σi
γi and E(G)= ( )i
.i
E G
In particular, for a 1-2-3-edge weighting and vertex coloring of complete graph Kn, we have three
partitions E(Kn)1, E(Kn)2 and E(Kn)3 as follows:
E(Kn)1={vivj∈E(Kn)|f(vivj)=1}⇒|E(Kn)1|=γn,
E(Kn)2={vivj
∈E(Kn)|f(vivj)=2}⇒|E(Kn)2|=βn,
E(Kn)3={vivj
∈E(Kn)|f(vivj)=3}⇒|E(Kn)3|=αn.
such that mn=|E(Kn)|=γ+β+α and E(Kn)1
∪ E(Kn)2
∪ E(Kn)3=E(Kn) (obviously, mn=( )2
( 1)
2
n n n −
= ).
Proof of Theorem 1.1. Consider the complete graph K3q ,q∀ ∈ with 3q vertices and 3
2
q(3q-
1) edges. For obtain all aims in Theorem 1.1, we present an algorithm for weighting all edges of
K3q with labels 1, 2 and 3.
2.1. ALGORITHM FOR 1-2-3-EDGE WEIGHTING AND VERTEX COLORING OF K3q (q ≥5):
1- Choose an arbitrary vertex v∈V(K3q) and label all inside edges to v with 3. If we name v by v1,
then ∀ u∈V(K3q) f(v1u)=3 and S1=3(3q-1).
2- Choose one of adjacent vertices with v1 (we name v2) and label all inside edges to v2 with 3;
except an edge v2u (that we name u by v3q), then for all u∈V(K3q), u≠v3q f(v2u)=3 and f(v2v3q)=2.
Thus S2=S1-1=9q-4.
3- Choose one of adjacent vertices with v1,v2 and name v3. We label all inside edges to v3 with 3;
except two edges v3v3q,v3u, then ∀ w∈V(K3q), w≠v3q,u f(v3w)=3 and f(v3v3q)=f(v3u)=2 (we name u
by v3q-1). Thus S3=S2-1=9q-5.
4- Choose one of adjacent vertices with v1,v2,v3 and name v4. We label all inside edges to v4 with
3; except two edges v4v3q,v4v3q-1 then ∀ w∈V(K3q), w≠v3q,v3q-1 f(v4w)=3, f(v4v3q-1)=2 and
f(v4v3q)=1. Thus S4=S3-1=9q-6.
I- (2≤I≤2q+1) Choose an arbitrary vertex v∈V(K3q) that didn't choose above (we name vi). So
If I be even, then we label all edge vivj with 3 for 1≤j≤3q-
[
2
]
1i +
, all edge vivj with 1 for 3q≥j≥3q-
[
2
]
1i +
+2 and label vivh (h=3q-
[
2
]
1i +
+1) with 2.
Else, I be odd, then we label all edge vivj with 3 for 1≤j≤3q-1-
[
2
]
1i +
, all edge vivj with 1 for
3q≥j≥3q-
[
2
]
1i +
+3 and label vivh (h=3q-
[
2
]
1i +
+1, 3q-
[
2
]
1i +
+2) with 2.
In other words,
3. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013
21
( )1 3q3q 3q+1 3q+2
2 2 2
2,...,2 1 ... ( ) 3, ( ) 2, ( ) ... ( ) 1.i ii i i i ii
i q f v v f v v f v v f v v f v v − − −
∀ = = = = = = = =
( )1 3q3q 3q 3q+1 3q+2
2 2 2 2
3,...,2 1 ... ( ) 3, ( ) ( ) 2, ( ) ... ( ) 1.i i ii i i i iiii q f v v f v v f v v f v v f v v f v v − − − −
∀ = + = = = = = = = =
And obviously, Si= Si-1-1=9q-2-i.
2Q+2- Choose one of adjacent vertices with v1,v2,…,v2q+1 and name v2q+2. Label two edges
v2q+2v3q-1 and v2q+2v3q with label 2, 1 respectively and other inside edges to v2q+2 with 3, that
haven't a label. Therefore S2q+2=S2q+1-3=7q-6.
J- (3q-2≥J≥2q+3=3q-(q-3)) Choose a remaining vertex and name vj. Now, let j=3q-h (q-3≥h≥2)
and label three edges vjvk for k=2h+1, 2h+2, 2h+3 with label 2. Also, label all edges vjvk with
label 3 for 1≤k≤2h=6q-2j and other edge vjvk give label 1 (6q-2j+4=2h+4≤k≤3q). Then obviously
Sj=Sj-1-4=7q-6-4(j-(2q+2)) and vertex-color of vj is Sj=15q+2-4j.
3Q-2- For v3q-2, there are f(v3q-2v1)=f(v3q-2v2)=f(v3q-2v3)=f(v3q-2v4)=3, f(v3q-2v5)=f(v3q-2v6)=f(v3q-
2v7)=2 and f(v3q-2v8)=…= f(v3q-2v3q)=1. Thus S3q-2=S3q-3-4=3q+10.
3Q-1- For v3q-1, there are f(v3q-1v1)=f(v3q-1v2)=3, f(v3q-1v3)= f(v3q-1v4)= f(v3q-1v5)=2 and f(v3q-
1v6)=…= f(v3q-1v3q)=1. Thus S3q-1=S3q-2-4=3q+6.
Figure 1. The example of a 1-2-3-edge weighting and vertex coloring of K3 and K6.
3Q- Finally, for vertex v3q, the edge v3qv1 labeled with 3 and v3qv2 and v3qv3 labeled with 2. Also
all edge v3qvi (i=4,…,3q-1) labeled with one. Thus S3q=S3q-1-3=3q+3.
It is obvious that by running this algorithm on complete graph K3q for q=5,6,…, we can obtain its
1-2-3-edge weighting and vertex coloring. For small number 1,2,3,4, reader can see following
figures. The 1-2-3-edge weighting and vertex coloring of K3, K6, K9 and K12 aren't taken some
steps of above algorithm. These graphs are shown in Figure 1 and Figure 2. Also, reader can see
the running algorithm on K15 in Figure 3. Thus, by obtaining two functions
The edge weighting f :E(K3q)→ {1,2,3} and the vertex coloring s:V(K3q)→S={9q-3,9q-4,…,7q-
2,7q-3,7q-6,7q-10,…,3q+10,3q+6,3q+3}, proof of Theorem 1.1 is completed. □
4. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013
22
Figure 2. Two example of algorithm to attain a 1-2-3-edge weighting and vertex coloring of K9 and K12.
Figure 3. An example of algorithm to attain a 1-2-3-edge weighting and vertex coloring of K15.
3. OPEN PROBLEMS AND CONJECTURES
In this section, we calculate some result that concluded from Theorem 1.1 and the algorithm for
1-2-3-edge weighting and vertex coloring of K3q. In continue, we introduce some open problem
and conjectures, which some of them maybe could be solved in the near future, by other reader.
Lemma 3.1. Consider the 1-2-3-edge weighting and vertex coloring of K3q= that obtained from
above algorithm and using denotations of Definition 2.1, we have γ3q=
2
3q 5q 2
,
2
− +
β3q=3q-2
and α3q=3q2
-2q+1.
Proof. The proving of lemma is easy; by refer to the algorithm for 1-2-3-edge weighting and
vertex coloring of K3q in proof of Theorem 1.1. Since obviously, q and 2q edges with weight 2 are
inside to the vertices v2,v4,…,v2q and v3,v6,…,v2q+1, respectively. Also, 3 edges with weight 2 are
inside the vertex vi (i=2q+2,…,3q-1) and two edges with weight 2 are inside to v3q. Thus
β3q=
2 3( 2) 2
2
q q q+ + − +
=3q-2.□
5. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013
23
Conjecture 3.2. (The 1-2-3-conjecture [6,8,10]) Every connected graph G=(V,E) non-isomorph
to K2 (with at least two edges) has an edge weighting f:E→{1,2,3} and vertex coloring s:V→{n-
1,...,3n-3}.
Conjecture 3.3. (n vertex coloring) There are distinct numbers of Sv's, v∈V(G) of a graph G of
order n, for a 1-2-3-edge weighting and vertex coloring.
Conjecture 3.4. (The 1,2-conjecture [7,11,12]) Every graph G has a coloring chip configuration
c:V ∪ E→{1,2}.
Conjecture 3.5. (Antimagic weighting [3,5]) For every connected graph G (with at least two
edges) there is a bijection c:E→{1,2,…,|E|} such that no two vertices of G have the same
potential.
Conjecture 3.6. (Proper vertex coloring) For all graph G of order n, there are χ(G) numbers of
Sv's, v∈V(G) with this 1-2-3-edge weighting and vertex coloring. Where χ(G) is the number
colors of the vertices on the graph G.
Conjecture 3.7. (Lucky weighting, [3,4]) For every graph G, there is a vertex weighting
c:V→{1,2,...,χ(G)}, whose vertex potential ( )
( )v u N v
q c u∈
=∑ is a proper coloring of G.
ACKNOWLEDGEMENTS
The authors are thankful to Dr. Mehdi Alaeiyan and Mr Hamid Hosseini of Department of
Mathematics, Iran University of Science and Technology (IUST) for their precious support and
suggestions.
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