Dirac-type Problem of Rainbow matchings and Hamilton cycles in Random Graphs

TL;DR

This paper proves that in random graphs, with high probability, large minimum degree spanning subgraphs guarantee the existence of rainbow Hamilton cycles and perfect matchings, extending Dirac-type results to rainbow settings.

Contribution

It establishes Dirac-type conditions ensuring rainbow Hamilton cycles and perfect matchings in random graphs, a novel extension of classical Hamiltonicity results.

Findings

Rainbow Hamilton cycles exist with high probability under specified degree conditions.

Rainbow perfect matchings are guaranteed in certain random graph families.

Results extend Dirac's theorem to rainbow structures in random graphs.

Abstract

Given a family of graphs $G_1,\dots,G_{n}$ on the same vertex set $[n]$, a rainbow Hamilton cycle is a Hamilton cycle on $[n]$ such that each $G_c$ contributes exactly one edge. We prove that if $G_1,\dots,G_{n}$ are independent samples of $G(n,p)$ on the same vertex set $[n]$, then for each $\varepsilon>0$, whp, every collection of spanning subgraphs $H_c\subseteq G_c$, with $\delta(H_c)\geq(\frac{1}{2}+\varepsilon)np$, admits a rainbow Hamilton cycle. A similar result is proved for rainbow perfect matchings in a family of $n/2$ graphs on the same vertex set $[n]$.