Proper edge colorings of planar graphs with rainbow C 4 ${C}_{4}$ -s

Abstract

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 $q_B\left(G\right)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B\left(G\right)$ for planar and outerplanar graphs in terms of the maximum degree $\Delta =\Delta \left(G\right)$. We prove that $q_B\left(G\right)\le 2\Delta +8$ for planar graphs, $q_B\left(G\right)\le 2\Delta$ for bipartite planar graphs, and $q_B\left(G\right)\le \Delta +1$ for outerplanar graphs with $\Delta \ge 4$. We conjecture that, for $\Delta$ sufficiently large, $q_B\left(G\right)\le 2\Delta \left(G\right)$ for planar $G$, and $q_B\left(G\right)\le \Delta \left(G\right)$ for outerplanar $G$.