On strong incidence coloring of k-degenerate graphs

An incidence of a graph G is a pair (u,e) where u is a vertex of G and e is an edge of G incident with u. Two incidences (u,e) and (v,f) of G are adjacent whenever (i) u = v, or (ii) e = f or (iii) uv = e or f. A strong incidence coloring of a graph G is a mapping from the set of incidences of G to the set of colors, such that every two incidences that are adjacent or adjacent to a same incidence receive distinct colors. The minimum number of colors needed for a strong incidence coloring of a graph is called the strong incidence chromatic number. In this paper, we prove that the strong incidence chromatic number of each k–degenerate graph G is at most 6k∆(G) − 3k2− 2∆(G) + 1.