 Any graph with a nonempty set of edges requires at least two colors; if "G" and "H" are not 1-colorable, that is, they both contain an edge, then their product also contains an edge, and is hence not 1-colorable either.In particular, the conjecture is true when "G" or "H" is a bipartite graph, since then its chromatic number is either 1 or 2.Similarly, if two graphs "G" and "H" are not 2-colorable, that is, not bipartite, then both contain a cycle of odd length.