Equitable oriented coloring

Abstract

AbstractIn this paper, we consider equitable oriented colorings of graphs. Such coloring is a natural combination of two well‐known colorings: oriented coloring and equitable coloring. An oriented ‐coloring of an oriented graph is an arc‐preserving homomorphism from into an oriented graph on vertices, or colors. We say that the coloring is equitable if for every two colors , we have . This paper presents the following results: we show that there is an undirected graph such that every orientation of is equitably oriented 5‐colorable, and we show that there is an orientation of that cannot be colored with six colors. In other words, there is a gap in the set of numbers of colors that can be used for equitable oriented coloring of . Additionally, we concentrate on paths and show that there is an orientation of a path that requires four colors to be equitably oriented colorable. On the other hand, we show that every orientation of any path is equitably oriented ‐colorable, for every .