A decomposition for total‐coloring partial‐grids and list‐total‐coloring outerplanar graphs

Abstract

AbstractThe total chromatic number χT(G) is the least number of colors sufficient to color the elements (vertices and edges) of a graph G in such a way that no incident or adjacent elements receive the same color. In the present work, we obtain two results on total‐coloring. First, we extend the set of partial‐grids classified with respect to the total‐chromatic number, by proving that every 8‐chordal partial‐grid of maximum degree 3 has total chromatic number 4. Second, we prove a result on list‐total‐coloring biconnected outerplanar graphs. If for each element x of a biconnected outerplanar graph G there exists a set Lx of colors such that |Luw| = max{deg(u) + 1, deg(w) + 1} for each edge uw and |Lv| = 7 − δdeg(v),3 − 2δdeg(v),2 (where δi,j = 1 if i = j and δi,j = 0 if i ≠ j) for each vertex v, then there is a total‐coloring π of graph G such that π(x) ∈ Lx for each element x of G. The technique used in these two results is a decomposition by a cutset of two adjacent vertices, whose properties are discussed in the article. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011