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

A {\em 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 {\em total chromatic number} of a graph $G$, denoted by \chiup''(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 $\Delta$ admits a total coloring with at most $\Delta + 2$ colors. A graph is {\em $1$-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 $\Delta$ at least~$11$ and without adjacent triangles, then $G$ admits a total coloring with at most $\Delta + 2$ colors.Comment: 10 page

On Chromatic Number of Colored Mixed Graphs

An (m, n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m, n)-colored mixed graph G to an (m, n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f (u)f (v) is an arc (edge) of color c in H. The (m,n)-colored mixed chromatic number x((m,n))(G) of an (m, n)-colored mixed graph G is the order (number of vertices) of a smallest homomorphic image of G. This notion was introduced by Nektfil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147-155). They showed that x((m,n))(G) <= k(2m n)(k-1) where G is a acyclic k-colorable graph. We prove the tightness of this bound. We also show that the acyclic chromatic number of a graph is bounded by k(2) k(2+) inverted left perpedicular log((2m+n)) log((2m+n)) k inverted right perpendicular if its (m, n)-colored mixed chromatic number is at most k. Furthermore, using probabilistic method, we show that for connected graphs with maximum degree its (m, n)-colored mixed chromatic number is at most 2(Delta-1)(2m+n) (2m + n)(Delta-min(2m+m3)+2) In particular, the last result directly improves the upper bound of 2 Delta(2)2(Delta) oriented chromatic number of graphs with maximum degree Delta, obtained by Kostochka et al. J. Graph Theory 24, 331-340) to 2(Delta-1)(2)2(Delta). We also show that there exists a connected graph with maximum degree Delta and (m, n)-colored mixed chromatic number at least (2m + n)(Delta/2)