On lengths of rainbow cycles

Abstract

We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge. We settle a question posed by Ball, Pultr, and Vojtěchovský by showing that if such a coloring does not contain a rainbow cycle of length $n$, where $n$ is odd, then it also does not contain a rainbow cycle of length $m$ for all $m$ greater than $2n^2$. In addition, we present two examples which demonstrate that this result does not hold for even $n$. Finally, we state several open problems in the area.

10 pages, 7 figures, 1 table. Replaced version (v2) includes a new result, some computer experimentation, and more. The next version (v3) is only to correct the title of the arXiv listing. The latest version (v4) makes several important changes, additions, and updates