Coloring, labeling andmapping of graph

Abir Naskar

10MA40001

Under the guidance of

Prof. Pawan Kumar

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by

abstract• In this content we will see that what will

be coloring of a graph G in generalized sense. We also see that the span of L(2,1) labeling when minimum and

maximum degree is given. And finally we will map a graph on a weighted graph

taken as a path and we will see what will be the minimum weight in each case.

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coloring

•Introduction•What is coloring ?

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resultsNumber 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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• 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 evenIf it be odd then replace k by k+1 and put in the previous formula will give the answer

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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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Petersen graph

For ∆=37

Modifying the previous graph we get

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For ∆=4

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∆=5

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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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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For minimum degree ∆≥3 Some graphs..

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resultsfor 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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Mapping of a graph on a path

• Introduction• What is mapping of a graph on another graph?

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• Let us first concentrate on following graph

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• 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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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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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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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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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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Thank you…

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