This document summarizes research on graph coloring, labeling, and mapping. It discusses coloring graphs with k colors, the number of possible colorings of cycles. It also covers L(2,1) labeling, where the span is minimized based on minimum degree. Finally, it examines mapping graphs onto paths, where the minimum total weight is found based on the number of vertices and diameter.

1. Coloring, labeling and
mapping of graph
Abir Naskar
10MA40001
Under the guidance of
Prof. Pawan Kumar
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by

2. abstract
• In thiscontent wewill seethat what will becoloring
of agraph G in generalized sense. Wealso seethat
thespan of L(2,1) labeling when minimum and
maximum degreeisgiven. And finally wewill map
agraph on aweighted graph taken asapath and we
will seewhat will betheminimum weight in each
case.
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3. coloring
• Introduction
• What is coloring ?
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4. results
Number of coloring of a cycle when k is given
If m ≤ 2k+1 then color is n
If m = 0(mod k+1) then color is k+1
If m ≠ 0(mod k+1) then color is k+2
Which will give the number of coloring is
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5. • Coloring of following graph
We get the number of color will be less or equal with (k + 1) + 2(k - 1) +2(k-3)+ ….. + 2
if k is even
If it be odd then replace k by k+1 and put in the previous formula will give the answer
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6. tHe l(2,1) labeling anD
oPeration oF graPH
• Introduction
• What is L(p,q) labeling?
• Conjecture (Griggs and Yeh)
for maximum degree ∆≥2,
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7. Petersen graph
For ∆=3
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8. Modifying the previous
graph we get
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For ∆=4

9. 9
∆=5

10. results
f(2) ≤ 4, f(3) ≤ 9 and
f(∆) ≤ f(∆-1)+∆ when ∆>3
• Result for L(k,k-1,…,1) labeling
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11. Let us first see the result
For a graph of minimum degree ∆ we have the
L(2,1) labeling with span
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12. For minimum degree∆≥3 Some
graphs..
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13. 13

14. results
for minimum degree ∆ the minimum number of coloring is
5 when ∆ =3
7 when ∆ =4 & 5
9 when ∆ =6 & 7
11 when ∆= 8 & 9
And so on…(graphically we can show that)
For odd ∆ we can show that the number of labeling is minimum, ∆+2 (follows from previous theorem)
For even ∆ we found that it is ∆+3
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15. Mapping of a graph on a path
• Introduction
• What is mapping of a graph on another graph?
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16. • Let us first concentrate on following graph
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17. • Now the mapping of that graph on a path will give the total weight as
(2k∆ - 2k) and in that case number of vertices will be n = k∆ + 1 and
then we can write the total weight as (2n - 2k - 2)
• For flaps like …
total number of vertices will be
and the total weight will be
which will be equal with
Which we can write as . now if the diameter of
the graph will be “d” then we have and we can write the
total weight as
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18. results
• Any tree can be mapped on ( n > 1) with weight (2n-3) or
less with any pair of adjacent vertices as end point
• any tree can be mapped on a path with the weight 2n-d-2 or
less as any pair of vertices at distance d as end points
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19. Mapping of k-flap star graph on
following graph
• 2(n-1)+k(∆-2)+(k+2)( ) if n is odd
• 2n+k(∆-2)+(k+2)+(k+2) if n is even
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20. Significance
• In this short note we tried to solve some short problems
which are used in research paper and I wish that it will
definitely helps those who works on this topic.
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21. referenceS
• L(2,1)-Labeling in the Context of Some Graph Operations by S K Vaidya
• On the L(2, 1)-labeling of block graphs by Flavia Bonomo and Marcia R. Cerioli
• The L(2,1)-Labeling Problem on Graphs by Gerard J. Chang and David Kuo
• L(2,1)-labelling of graphs by Frederic Havet , Bruce Reed and Jean-Sebastien Sereni
• Graph theory by Douglas B. West chapter 5 section 5.1 and 5.2
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22. Thank you…
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