Rainbow odd cycles
Ron Aharoni, Joseph Briggs, Ron Holzman, Zilin Jiang

TL;DR
This paper proves that in complete graphs, any family of odd cycles necessarily contains a rainbow odd cycle, and characterizes families that lack such cycles, advancing understanding of rainbow cycle structures.
Contribution
It establishes a universal existence result for rainbow odd cycles in complete graphs and characterizes families of cycles without rainbow cycles.
Findings
Every family of odd cycles in $K_n$ has a rainbow odd cycle.
Characterization of families in $K_{n+1}$ without rainbow odd cycles.
Complete classification of cycle families without rainbow cycles.
Abstract
We prove that every family of (not necessarily distinct) odd cycles in the complete graph on vertices has a rainbow odd cycle (that is, a set of edges from distinct 's, forming an odd cycle). As part of the proof, we characterize those families of odd cycles in that do not have any rainbow odd cycle. We also characterize those families of cycles in , as well as those of edge-disjoint nonempty subgraphs of , without any rainbow cycle.
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