Hamiltonicity of graphs perturbed by a random regular graph

Abstract

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

Hamiltonicity of graphs perturbed by a random geometric graph

“…To the best of our knowledge, all of these results consider (hyper/di)graphs perturbed by a binomial random structure, such as $${G}_{n,p}$$, or its $${G}_{n,m}$$ counterpart. Only very recently, Espuny Díaz and Girão [12] considered Hamiltonicity in graphs perturbed by a random regular graph.…”