Proper edge colorings of planar graphs with rainbow $C_4$-s

TL;DR

This paper investigates the minimum number of colors needed for proper edge colorings of planar and outerplanar graphs where every 4-cycle is rainbow-colored, providing upper bounds related to maximum degree and proposing conjectures for large degrees.

Contribution

It establishes new upper bounds on the number of colors for B-colorings in planar and outerplanar graphs, advancing understanding of rainbow 4-cycle colorings in these classes.

Findings

q_B(G) ≤ 2Δ + 8 for planar graphs

q_B(G) ≤ 2Δ for bipartite planar graphs

q_B(G) ≤ Δ + 1 for outerplanar graphs with Δ ≥ 4

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 $G$.