Some New Results on the Adjacent Vertex Distinguishing Total Coloring

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1 1 / 9 Some New Results on the Adjacent Vertex Distinguishing Total Coloring Chao Fugang Qiang Huiying and Zhang Zhongfu Institute of Applied Mathematics Lanzhou Jiaotong University Lanzhou, Chaofugang2008@yahoo.com.cn

2 2 / 9 Main Contents 1 Abstract 2 3 Unsolved problems,open problems and Conjectures 4 References

3 3 / 9 Abstract Abstract An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of a graph G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denotes by χ at (G) is called adjacent vertex distinguishing total chromatic number.in this paper,we prove χ at (G) 5 for such graph with maximum degree (G) = 2 and χ at (G) 7 for such graph with maximum degree (G) = 4,we also introduce some recent results on adjacent vertex distinguishing total coloring,unsolved problems,open problems and conjectures.

4 4 / 9 Lemma 1 (zhang et al,[1]) If graph G has k components G 1, G 2,, G k, and V(G i ) 2, i = 1, 2,, k, then χ at (G) = max{χ at (G 1 ), χ at (G 2 ),, χ at (G k )}. Lemma 2 (zhang et al,[1]) Let C n be a cycle with order n,n 4,then χ at (C n ) = 4.

5 4 / 9 Lemma 1 (zhang et al,[1]) If graph G has k components G 1, G 2,, G k, and V(G i ) 2, i = 1, 2,, k, then χ at (G) = max{χ at (G 1 ), χ at (G 2 ),, χ at (G k )}. Lemma 2 (zhang et al,[1]) Let C n be a cycle with order n,n 4,then χ at (C n ) = 4.

6 5 / 9 Theorem 3 For any graph G with (G) = 2,we have χ at (G) 5. Theorem 4 (Wang,[2]) For any graph G with (G) = 3,we have χ at (G) 6.

7 5 / 9 Theorem 3 For any graph G with (G) = 2,we have χ at (G) 5. Theorem 4 (Wang,[2]) For any graph G with (G) = 3,we have χ at (G) 6.

8 Lemma 5 (Petersen,1891) If G is 2n-regular multigraph for some n 1, then G is 2-factorizable. Lemma 6 Let ϕ be a (proper) vertex-coloring of a cycle C n.let C be a set of colours such that C = 4. Then ϕ can be extended to an adjacent vertex distinguishing total coloring of C n such that ϕ(e) C for any edge of C n. 6 / 9

9 Lemma 5 (Petersen,1891) If G is 2n-regular multigraph for some n 1, then G is 2-factorizable. Lemma 6 Let ϕ be a (proper) vertex-coloring of a cycle C n.let C be a set of colours such that C = 4. Then ϕ can be extended to an adjacent vertex distinguishing total coloring of C n such that ϕ(e) C for any edge of C n. 6 / 9

10 7 / 9 Theorem 7 For any 4-regular graph G,we have χ at (G) 7.

11 8 / 9 Unsolved problems,open problems and Conjectures Unsolved problems,open problems and Conjectures Conjecture(zhang et al,[1]): For any connected simple graph G with order at least 2,we have χ at (G) (G) + 3. Unsolved problems: 1.graph with low degree and high degree;2.bound on an adjacent vertex distinguishing total coloring;3.random graph. Open problem(zhang et al,[1]):h is subgraph of G,when χ at (H) χ at (H).

12 9 / 9 References References 1.Zhang Zhongfu,ChenXiang en et al,on adjacent vertex distinguishing total coloring of graphs,science in China Series A:Mathematics 2005Vol.48No.3 : Wang Haiying and Sun Liang,Adjacent vertex distinguishing total chromatic number of graphs with maximum degree 3,to appear. 3.H.P Yap,Total coloring of graphs,springer,1996.