On incidence coloring and star arboricity of graphs

In this note we show that the concept of incidence coloring introduced in [BM] is a special case of directed star arboricity, introduced in [AA]. A conjecture in [BM] concerning asmyptotics of the incidence coloring number is solved in the negative following an example in [AA]. We generalize Theorem 2.1 of [AMR] concerning the star arboricity of graphs to the directed case by a slight modification of their proof, to give the same asymptotic bound as in the undirected case. As a result, we get tight asymptotic bounds for the maximum incidence coloring number of a graph in terms of its degree. 1 Connection between star arboricity and incidence coloring A star forest is a graph whose connected components are stars. A directed star forest is a graph whose connected components are directed stars (edges are directed out of the center). The star arboricity of an undirected graph G (introduced in [AK]) denoted st(G) is the smallest number of star forests needed to cover G. For a directed graph D, the directed star arboricity denoted dst(D), is the smallest number of directed star forests needed to cover D. In [BM], the concept of incidence coloring of graphs is introduced. ι(G) is the smallest number of colors needed to color the incidences (incident vertex-edge pairs) of an undirected graph G so that neighborly incidences do not receive the same color. Two incidences (v, e) and (w, f) are said to be neighborly if (i) v = w, (ii) e = f , or (iii) {v, w} is one of the edges e, f . If we think of an incidence pair as a directed edge, directed toward the vertex, we are coloring the edges of the symmetrically directed graph S(G) (we replace