Proper edge colorings of planar graphs with rainbow $C_4$-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 $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B(G)$ for planar and outerplanar graphs in terms of the maximum degree $\Delta =\Delta (G)$. We prove that $q_B(G)\le 2\Delta +8$ for planar graphs, $q_B(G)\le 2\Delta$ for bipartite planar graphs and $q_B(G)\le \Delta +1$ for outerplanar graphs with $\Delta \ge 4$. We conjecture that, for $\Delta$ sufficiently large, $q_B(G)\le 2\Delta (G)$ for planar $G$ and $q_B(G)\le \Delta (G)$ for outerplanar $G$.

The previous version of this manuscript had an error in the proof of Theorem 1.4(i). This error is corrected in the new version

References in corpus (1)

• A survey on star edge-coloring of graphs