On the oriented coloring of the disjoint union of graphs

Abstract

Let G⃗ = (V, A) be an oriented graph and G the underlying graph of G⃗. An oriented k-coloring of G⃗ is a partition of V into k color classes, such that there is no pair of adjacent vertices belonging to the same class and all the arcs between a pair of color classes have the same orientation. The smallest k such that G⃗ admits an oriented k-coloring is the oriented chromatic number χo(G⃗) = k of G⃗. The oriented chromatic number χo(G) of the undirected graph G is the maximum of χo(G⃗) for all orientations G⃗ of G. Oriented chromatic number of the product of two graphs G1, G2 was widely studied, but the disjoint union G1 ∪ G2 has not yet been considered. In this article we proved bounds for the oriented chromatic number of any two oriented graphs and we also proved that given two complete graphs Kn and Km with n ≥ m, there is a real number α ∈ (1, 3) such that χo(Kn ∪ Km) = n + m − α log2(m). Additionally, we established exact values of the union of one complete graph with one cycle and of one complete graph with a forest.