On a rainbow version of Dirac's theorem

TL;DR

This paper proves the existence of rainbow Hamilton cycles and perfect matchings in collections of graphs with high minimum degree, confirming a conjecture and extending Dirac's theorem to a rainbow setting.

Contribution

It establishes the existence of rainbow Hamilton cycles and perfect matchings under certain degree conditions, confirming a conjecture of Aharoni.

Findings

Existence of rainbow Hamilton cycles under minimum degree conditions

Existence of rainbow perfect matchings under similar conditions

Extension of Dirac's theorem to rainbow graph collections

Abstract

For a collection $\mathbf{G}=\{G_1,\dots, G_s\}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $\mathbf{G}$-transversal if there exists a bijection $\phi:E(H)\rightarrow [s]$ such that $e\in E(G_{\phi(e)})$ for all $e\in E(H)$. We prove that for $|V|=s\geq 3$ and $\delta(G_i)\geq s/2$ for each $i\in [s]$, there exists a $\mathbf{G}$-transversal that is a Hamilton cycle. This confirms a conjecture of Aharoni. We also prove an analogous result for perfect matchings.