1factorizations of pseudorandom graphs
Abstract
A $1$factorization of a graph $G$ is a collection of edgedisjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$factorization is that $V(G)$ is even and $G$ is regular; the converse is easily seen to be false. In this paper, we consider the problem of finding $1$factorizations of regular, pseudorandom graphs. Specifically, we prove that an $(n,d,\lambda)$graph $G$ (that is, a $d$regular graph on $n$ vertices whose second largest eigenvalue in absolute value is at most $\lambda$) admits a $1$factorization provided that $n$ is even, $C_0\leq d\leq n1$ (where $C_0$ is a universal constant), and $\lambda\leq d^{1o(1)}$. In particular, since (as is well known) a typical random $d$regular graph $G_{n,d}$ is such a graph, we obtain the existence of a $1$factorization in a typical $G_{n,d}$ for all $C_0\leq d\leq n1$, thereby extending to all possible values of $d$ results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed $d$. Moreover, we also obtain a lower bound for the number of distinct $1$factorizations of such graphs $G$ which is off by a factor of $2$ in the base of the exponent from the known upper bound. This lower bound is better by a factor of $2^{nd/2}$ than the previously best known lower bounds, even in the simplest case where $G$ is the complete graph. Our proofs are probabilistic and can be easily turned into polynomial time (randomized) algorithms.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.10361
 Bibcode:
 2018arXiv180310361F
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics