On the number of factorable induced subgraphs

Abstract

Let $F$ be an $r$-vertex graph. In this paper, we study the $F$-factor problem in random induced subgraphs of dense graphs. We show that for any $r$-vertex graph $F$ and $\gamma >0$, if $H$ is an $n$-vertex graph with minimum degree at least $(1-1/{\chi }_{cr}(F)+\gamma )n$, then for every fixed $p\in (0,1)$, the random induced subgraph $H[p]$ contains an $F$-factor with probability at least $1/(rq)-o_n(1)$, where $q\in \mathbb{N}$ is the order of certain coset group defined from $H$. The probability is asymptotically best possible for infinitely many $F$ and $H$ and yields that a $1/(rq)-o_n(1)$ proportion of the subsets of $H$ induce $F$-factors, interestingly, regardless of whether $H$ itself admits an $F$-factor. Similar results are obtained for perfect matchings in hypergraphs under minimum degree conditions. Our proof combines concentration inequalities, lattice point counting in ${\mathbb{Z}}^d$ and structural theorems for $F$-factors in dense (hyper)graphs.

20 pages