On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons
Matt Noble

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 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 , at least two edges of that cycle receive the same color. A graph is defined to be \textit{PRCF-good} if it admits a PRCF edge coloring, and 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
