A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least...A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without 1 cycles is at most △（G） ＋ 2 if （k, l） ∈ {（6, 3）, （7, 4）, （6, 5）, （7, 6）}.展开更多
基金Supported by.National Natural Science Foundation of China （Grant Nos. 10971121, 10631070, 60673059）Acknowledgements We would like to thank the referees for providing some very helpful suggestions for revising this paper.
文摘A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without 1 cycles is at most △（G） ＋ 2 if （k, l） ∈ {（6, 3）, （7, 4）, （6, 5）, （7, 6）}.
