Hamiltonicity of graphs perturbed by a random regular graph

TL;DR

This paper investigates how adding a small random regular graph to a dense deterministic graph affects Hamiltonicity and pancyclicity, providing probabilistic results and polynomial algorithms for cycle detection.

Contribution

It establishes new probabilistic thresholds for pancyclicity in graphs combined with random 1- and 2-regular graphs, extending previous results and offering efficient cycle-finding algorithms.

Findings

H∪G is pancyclic with high probability for 2-regular G and all α∈(0,1].

H∪G is pancyclic with high probability for 1-regular G and α∈(√2−1,1].

Polynomial-time algorithms are provided for finding cycles of any length.

Abstract

We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $\delta(H)\geq\alpha n$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $\alpha\in(0,1]$, and also extend our result to a range of sublinear degrees. When $G$ is a random $1$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $\alpha\in(\sqrt{2}-1,1]$, and this result is best possible. Furthermore, we show that this bound on $\delta(H)$ is only needed when $H$ is `far' from containing a perfect matching, as otherwise we can show results analogous to those of random $2$-regular graphs. Our proofs provide polynomial-time algorithms to find cycles of any length.