Proper Edge Colorings of Cartesian Products with Rainbow $$C_4$$-s

Abstract

AbstractRelated 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 $$q_B(G)$$ q B ( G ) denote the smallest number of colors needed for a B-coloring of G. Here we look at $$q_B(G)$$ q B ( G ) for Cartesian products of paths and cycles. Our main result is that $$q_B(G)$$ 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 $$q_B(G\square H)\le q_B(G)+q_B(H)$$ q B ( G □ H ) ≤ q B ( G ) + q B ( H ) if $$\chi (G)\le q_B(H), \chi (H)\le q_B(G)$$ χ ( G ) ≤ q B ( H ) , χ ( H ) ≤ q B ( G ) , where $$\chi (G)$$ χ ( G ) is the chromatic number of G.