Arc Colorings of Digraphs

A pair of arcs of a digraph D are consecutive if the terminal point of one is the initial point of the other. An arc-coloring of D is an assignment of colors to the arcs so that no pair of consecutive arcs have the same color and the arc-chromatic number, c(D), is the minimum number of colors in an arc-coloring of D. It is shown that if Tn is the transitive tournament on n points then c(Tn) = {log2n} but [((n + 1)2] colors suffice if the color classes are required to be oriented trees. It is further shown that if D is the complete digraph (an arc from any point to any other point) on n points then c(D) ∼ log2n. Finally it is shown that if a digraph D is n-arc-colorable it is 2n(point) colorable and this bound is best.