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{\bf Boris Alexeev}
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{\bf On Lengths of Rainbow Cycles}
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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 Vojt\v{e}ch\-ovsk\'{y} 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 a similar result does not
hold for even $n$. Finally, we state several open problems in the
area.
\bye