AbstractA biclique of G is a maximal set of vertices that induces a complete bipartite subgraph 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 . 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.