Total colorings of planar graphs with sparse triangles

The total chromatic number of a graph G, denoted by @g^''(G), is the minimum number of colors needed to color the vertices and edges of G such that no two adjacent or incident elements get the same color. It is known that if a planar graph G has maximum degree @D>=9, then @g^''(G)[email protected]+1. In this paper, we prove that if G is a planar graph with maximum degree 8, and for every vertex v, v is incident with at most d(v)[email protected]?d(v)[email protected]? triangles, then @g^''(G)=9.

[1]  Yue Zhao,et al.  On total 9-coloring planar graphs of maximum degree seven , 1999, J. Graph Theory.

[2]  Bing Wang,et al.  Total colorings of planar graphs without intersecting 5-cycles , 2012, Discret. Appl. Math..

[3]  Alexandr V. Kostochka,et al.  The total chromatic number of any multigraph with maximum degree five is at most seven , 1996, Discret. Math..

[4]  Oleg V. Borodin,et al.  On the total coloring of planar graphs. , 1989 .

[5]  Guiying Yan,et al.  Planar graphs with maximum degree 8 and without intersecting chordal 4-cycles are 9-totally colorable , 2012 .

[6]  Jian Chang,et al.  Total colorings of planar graphs with maximum degree 8 and without 5-cycles with two chords , 2013, Theor. Comput. Sci..

[7]  Yingqian Wang,et al.  On the 7 Total Colorability of Planar Graphs with Maximum Degree 6 and without 4-cycles , 2009, Graphs Comb..

[8]  Jean-Sébastien Sereni,et al.  Total colouring of plane graphs with maximum degree nine , 2007 .

[9]  Bing Wang,et al.  Total colorings of planar graphs with maximum degree seven and without intersecting 3-cycles , 2011, Discret. Math..

[10]  V. G. Vizing.  SOME UNSOLVED PROBLEMS IN GRAPH THEORY , 1968 .

[11]  Gerard J. Chang,et al.  Local condition for planar graphs of maximum degree 7 to be 8-totally colorable , 2011, Discret. Appl. Math..

[12]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[13]  Alexandr V. Kostochka,et al.  Total colorings of planar graphs with large maximum degree , 1997, J. Graph Theory.

[14]  Yingqian Wang,et al.  Total colorings of planar graphs with maximum degree at least 8 , 2009 .

[15]  Yingqian Wang,et al.  Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable , 2009, Discret. Appl. Math..