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An edge coloring of a simple graph $G$ is said to be \textit{properrainbow-cycle-forbidding} (PRCF, for short) if no two incident edges receivethe same color and for any cycle in $G$, at least two edges of that cyclereceive the same color. A graph $G$ is defined to be \textit{PRCF-good} if itadmits a PRCF edge coloring, and $G$ is deemed \textit{PRCF-bad} otherwise. Inrecent work, Hoffman, et al. study PRCF edge colorings and find many examplesof PRCF-bad graphs having girth less than or equal to 4. They then ask whethersuch graphs exist having girth greater than 4. In our work, we give astraightforward counting argument showing that the Hoffman-Singleton graphanswers this question in the affirmative for the case of girth 5. It is thenshown that certain generalized polygons, constructed of sufficiently largeorder, are also PRCF-bad, thus proving the existence of PRCF-bad graphs ofgirth 6, 8, 12, and 16.

On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons