On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

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