On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

Authors

Matt Noble, Middle Georgia State UniversityFollow

Publication Date

January 2023

Abstract

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 for any cycle in G, at least two edges of that cycle receive the same color. A graph G is defined to be PRCF-good if it admits a PRCF edge coloring, and G is deemed 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.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Noble, Matt (2023) "On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons," Theory and Applications of Graphs: Vol. 10: Iss. 1, Article 2.
DOI: 10.20429/tag.2022.100102
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol10/iss1/2