Rainbow even cycles

TL;DR

This paper proves that a large family of even cycles on a fixed set always contains a rainbow even cycle, resolving an open problem and establishing the result as optimal for all positive integers.

Contribution

It establishes the existence of a rainbow even cycle in any sufficiently large family of even cycles, solving a previously open problem in combinatorics.

Findings

Every family of at least loor 1.2(n-1)  even cycles contains a rainbow even cycle.

The result is proven to be optimal for all positive integers n.

The paper resolves an open problem posed by Aharoni, Briggs, Holzman, and Jiang.

Abstract

We prove that every family of (not necessarily distinct) even cycles $D_1, \dotsc, D_{\lfloor 1.2(n-1) \rfloor+1}$ on some fixed $n$-vertex set has a rainbow even cycle (that is, a set of edges from distinct $D_i$'s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, the result is best possible for every positive integer $n$.