Total colorings of degenerate graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree <= s. We prove that an s-degenerate graph G has a total coloring with Delta+1 colors if the maximum degree Delta of G is sufficiently large, say Delta >= 4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a linear-time algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.