On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

TL;DR

This paper investigates the existence of graphs with specific coloring properties, demonstrating that certain large generalized polygons and the Hoffman-Singleton graph are PRCF-bad, thus expanding known examples beyond girth 4.

Contribution

The paper proves the existence of PRCF-bad graphs with girth greater than 4, including the Hoffman-Singleton graph and large generalized polygons, using counting arguments and structural analysis.

Findings

Hoffman-Singleton graph is PRCF-bad with girth 5.

Certain large generalized polygons are PRCF-bad with girth 6, 8, 12, and 16.

PRCF-bad graphs exist beyond girth 4, contrary to previous beliefs.

Abstract

An edge coloring of a simple graph $G$ is said to be \textit{proper rainbow-cycle-forbidding} (PRCF, for short) if no two incident edges receive the same color and for any cycle in $G$, at least two edges of that cycle receive the same color. A graph $G$ is defined to be \textit{PRCF-good} if it admits a PRCF edge coloring, and $G$ is deemed \textit{PRCF-bad} otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than 4. In our work, we give a straightforward counting argument showing that the Hoffman-Singleton graph answers this question in the affirmative for the case of girth 5. It is then shown that certain generalized polygons, constructed of sufficiently large order, are also PRCF-bad, thus proving the existence of PRCF-bad graphs of girth 6, 8, 12, and 16.