Rainbow odd cycles

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 $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.