Total colorings of degenerated graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. A graph G is s-degenerated for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree less than or equal to s. We prove that an s-degenerated graph G has a total coloring with Delta + 1 colors if the maximum degree Delta of G is sufficiently large, say Delta greater than or equal to 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, i.e. the tree-width of G is bounded by a fixed integer k.