# On lengths of rainbow cycles

## by Boris Alexeev

### 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.

### Approximate citation

Boris Alexeev
On lengths of rainbow cycles
Electronic Journal of Combinatorics 13 (2006), no. 1, R105, 14 pp.