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On total chromatic number of planar graphs without 4-cycles

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Science China Mathematics 2007年第1期50卷 81-86页

作者： Min-le SHANGGUAN College of Mathematics Physics and Information Engineering Zhejiang Normal University

Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (total chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ? 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the total chromatic Conjecture.

关键词： total chromatic number planar graph F 5-subgraph 05C40

THE total chromatic number OF PSEUDO-OUTERPLANAR GRAPHS

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Applied Mathematics(A Journal of Chinese Universities) 1997年第4期12卷 455-462页

作者： WANGWEIFAN ZHANGKEMIN Departmentofmatbematics nanjinguniversitynanjing210093

A Planar graph g is called a ipseudo outerplanar graph if there is a subset v.∈V(G),[V.]=i,such that G-V. is an outerplanar graph in particular when *** a forest ,g is called a i-pseudo-tree .in this *** following results are proved;(1)the conjecture on the total coloring is true for all 1-pseudo-outerplanar graphs;(2)X1(G)+1 fo any 1-pseudo outerplanar graph g with △(G)≥3,where x4(G)is the total chromatic number of a graph g.

关键词： total chromatic number planar graph

total chromatic number of the Join of K_(m,n) and C_n

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Chinese Quarterly Journal of Mathematics 2006年第2期21卷 264-270页

作者： LI Guang-rong ZHANG Li-min College of Information Engineering Information Engineering Institute Zhengzhou 450002 China

The total chromatic number xT（G） of a graph G is the minimum number of colors needed to color the elements（vertices and edges） of G such that no adjacent or incident pair of elements receive the same color, G is called Type 1 if xT（G） =△（G）＋1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.

关键词： total coloring total chromatic number join graphs cycle complete bipartite graph

On Some total chromatic number ofOuterplanar Graphs with △(G) =6

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应用数学 1996年第4期 529-529页

作者： Zhang Zhongfu Lu Xinzhong Han Jincang Lin Kerong(Lanzhou Railway Institute,Lanzhou,730070)(Fuzhou Univ.,Fuzhou,350002)

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关键词： Graphs Outerplanar graphs total chromatic number

total Coloring of G×P_n and G×C_n

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The Journal of China Universities of Posts and Telecommunications 2000年第1期7卷 13-15页

作者： YANG Yi-xian, LIU Huan-ping (Information Security Center, Department of Information Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, P. R. China ) Information Security Center Department of Information Engineering Beijing University of Posts and Telecommunications Beijing 100876 P. R. China

It is proved that if G is a (+1)-colorable graph, so are the graphs G×Pn and C×Cn, where Pn and Cn are respectively the path and cycle with n vertices, and the maximum edge degree of the graph. The exact chromatic numbers of the product graphs and are also presented. Thus the total coloring conjecture is proved to be true for many other graphs.

关键词： combinatorial problems product graph total coloring total chromatic number