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Keywords:

  • total-coloring;
  • list-total-coloring;
  • graph decompositions;
  • partial-grids;
  • outerplanar graphs

Abstract

The 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 ij) 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