Universality of random graphs for graphs of maximum degree two

TL;DR

This paper establishes that random graphs with appropriate edge probability are universal for all graphs with maximum degree two, completing the understanding of universality thresholds across degrees.

Contribution

It proves that for maximum degree two graphs, the random graph $G(n,p)$ is universal with high probability at the optimal edge probability threshold.

Findings

Random graph $G(n,p)$ is universal for degree two graphs at $p ig(rac{ ext{log } n}{n}ig)^{1/2}$

Completes the universality threshold results for all degrees $d eq 1$

Supports the conjecture on universality thresholds for sparse random graphs

Abstract

For a family $\mathcal{F}$ of graphs, a graph $G$ is called \emph{$\mathcal{F}$-universal} if $G$ contains every graph in $\mathcal{F}$ as a subgraph. Let $\mathcal{F}_n(d)$ be the family of all graphs on $n$ vertices with maximum degree at most $d$. Dellamonica, Kohayakawa, R\"odl and Ruci\'nski showed that, for $d\geq 3$, the random graph $G(n,p)$ is $\mathcal{F}_n(d)$-universal with high probability provided $p\geq C\big(\frac{\log n}{n}\big)^{1/d}$ for a sufficiently large constant $C=C(d)$. In this paper we prove the missing part of the result, that is, the random graph $G(n,p)$ is $\mathcal{F}_n(2)$-universal with high probability provided $p\geq C\big(\frac{\log n}{n}\big)^{1/2}$ for a sufficiently large constant $C$.