Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles

A total coloring of a graph G is an assignment of colors to the vertices and the edges of G such that every pair of adjacent/incident elements receive distinct colors. The total chromatic number of a graph G, denoted by $$\chi ''(G)$$χ′′(G), is the minimum number of colors in a total coloring of G. The well-known total coloring conjecture (TCC) says that every graph with maximum degree $$\Delta $$Δ admits a total coloring with at most $$\Delta + 2$$Δ+2 colors. A graph is 1-toroidal if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 1-toroidal graphs, and prove that the TCC holds for the 1-toroidal graphs with maximum degree at least 11 and some restrictions on the triangles. Consequently, if G is a 1-toroidal graph with maximum degree $$\Delta $$Δ at least 11 and without adjacent triangles, then G admits a total coloring with at most $$\Delta + 2$$Δ+2 colors.

[1]  Jianfeng Hou,et al.  On total colorings of 1-planar graphs , 2013, J. Comb. Optim..

[2]  Xin Zhang,et al.  On edge colorings of 1-planar graphs , 2011, Inf. Process. Lett..

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

[4]  Guizhen Liu,et al.  On edge colorings of 1-planar graphs without adjacent triangles , 2012, Inf. Process. Lett..

[5]  Oleg V. Borodin.  A new proof of the 6 color theorem , 1995, J. Graph Theory.

[6]  H. Yap.  Total Colourings of Graphs , 1996 .

[7]  M. Rosenfeld,et al.  On the total coloring of certain graphs , 1971 .

[8]  J. Pach,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[9]  Janos Galambos.  A Probabilistic Approach to mean Values of Multiplicative Functions , 1970 .

[10]  D. de Werra,et al.  Graph Coloring Problems , 2013 .

[11]  André Raspaud,et al.  Acyclic colouring of 1-planar graphs , 2001, Discret. Appl. Math..

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

[13]  N. Vijayaditya.  On Total Chromatic Number of a Graph , 1971 .

[14]  Alexandr V. Kostochka,et al.  The total coloring of a multigraph with maximal degree 4 , 1977, Discret. Math..

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

[16]  G. Ringel.  Ein Sechsfarbenproblem auf der Kugel , 1965 .

[17]  Xin Zhang,et al.  List edge and list total coloring of 1-planar graphs , 2012 .