Abstract:

The arc-chromatic number of a digraph is the smallest number of colors required in an arc-coloring such that no two consecutive arcs get the same color. We first introduce some results on the arc-coloring of digraphs, then give a discussion about the upper bound of the arc-chromatic number of tournaments. By determining the maximum arc-chromatic number of tournaments with order at most $8$, we show that although the upper bound of the arc-chromatic number of general digraphs is tight, it is not tight for tournaments.

Key words: arc-coloring, arc-chromatic number, tournaments