Proper Edge Colorings of Cartesian Products with Rainbow C_4-s

Abstract

Related to the famous (7,4)-problem, in an earlier paper we introduced B-colorings. We call a proper edge coloring of a graph G a B-coloring if every 4-cycle of G is colored with four different colors. Let q_B(G) denote the smallest number of colors needed for a B-coloring of G. Here we look at q_B(G) for Cartesian products of paths and cycles. Our main result is that q_B(G) is equal to the chromatic index of G for grids, i.e. for Cartesian products of paths (apart from a few exceptions). This extends an earlier result for the case when G is the d-dimensional cube. Our main tool is a lemma which gives q_B(Gsquare H)le q_B(G)+q_B(H) if chi (G)le q_B(H), chi (H)le q_B(G), where chi (G) is the chromatic number of G.