Finding any given 2-factor in sparse pseudorandom graphs efficiently

Abstract

Given an $n$-vertex pseudorandom graph $G$ and an $n$-vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, i.e.\ an embedding $\ddot{I}†:V(H)\to V(G)$ so that $\ddot{I}†(u)\ddot{I}†(v)\in E(G)$ for all $uv\in E(H)$. Particular instances of this problem include finding a triangle-factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $(p,\hat{I}»)$-bijumbled graphs of minimum degree which is a constant proportion of the average degree, i.e.\ $\hat{I}\text{c◯}(pn)$. A $(p,\hat{I}»)$-bijumbled graph is characterised through the discrepancy property: $\left|e(A,B)-p|A||B|\right|<\hat{I}»\sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $\hat{I}»=O(p^{2}n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption-reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work (\emph{European Journal of Combinatorics} \textbf{82} (2019), 102999), incorporating the work of Nenadov (\emph{Bulletin of the London Mathematical Society} \textbf{51} (3) (2019), pp.~421--430), together with additional ideas and simplifications.

21 pages, final version