On star and biclique edge-colorings

Abstract

A biclique of G is a maximal set of vertices that induces a complete bipartite subgraph Kp,q of G with at least one edge, and a star of a graph G is a maximal set of vertices that induces a complete bipartite graph K1,q. A biclique (resp. star) edge-coloring is a coloring of the edges of a graph with no monochromatic bicliques (resp. stars). We prove that the problem of determining whether a graph G has a biclique (resp. star) edge-coloring using two colors is NP-hard. Furthermore, we describe polynomial time algorithms for the problem in restricted classes: K3-free graphs, chordal bipartite graphs, powers of paths, and powers of cycles.