A 1-factor in an n-vertex graph G is a collection of 
$$\frac{n}{2}$$
 vertex-disjoint edges and a 1-factorization of G is a partition of its edges into edge-disjoint 1-factors. Clearly, a 1-factorization of G cannot exist unless n is even and G is regular (that is, all vertices are of the same degree). The problem of finding 1-factorizations in graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Deciding whether a graph has a 1-factorization is usually a very difficult question. For example, it took more than 60 years and an impressive tour de force of Csaba, Kuhn, Lo, Osthus and Treglown to prove an old conjecture of Dirac from the 1950s, which says that every d-regular graph on n vertices contains a 1-factorization, provided that n is even and 
$$d \geqslant 2\left[ {\frac{n}{4}} \right] - 1$$
. In this paper we address the natural question of estimating F(n, d), the number of 1-factorizations in d-regular graphs on an even number of vertices, provided that 
$$d \geqslant \left[ {\frac{n}{2}} \right] + \varepsilon n$$
. Improving upon a recent result of Ferber and Jain, which itself improved upon a result of Cameron from the 1970s, we show that 
$$F\left( {n,\,d} \right) \geqslant {\left( {\left( {1 + o\left( 1 \right)} \right)\frac{d}{{{e^2}}}} \right)^{nd/2}}$$
, which is asymptotically best possible.

Number of 1-Factorizations of Regular High-Degree Graphs

Threshold for Steiner triple systems

A $1$-factorization of a graph $G$ is a collection of edge-disjoint 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, $C_0\leq d\leq n-1$ (where $C_0$ is a universal constant), and $\lambda\leq d^{1-o(1)}$. 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 $C_0\leq d\leq n-1$, 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.

/pdf/1-factorizations-of-pseudorandom-graphs-2939qf6yce.pdf

1-Factorizations of pseudorandom graphs

Inspired by an old result by Georg Frobenius, we show that the unbiased version of Hall’s marriage theorem is more transparent when reformulated in the language of matrices. At the same time, we ob...

Unbiased Version of Hall’s Marriage Theorem in Matrix Form

A $1$-factor in an $n$-vertex graph $G$ is a collection $\frac{n}{2}$ vertex-disjoint edges and a $1$-factorization of $G$ is a partition of its edges into edge-disjoint $1$-factors. Clearly, a $1$-factorization of $G$ cannot exist unless $n$ is even and $G$ is regular (that is, all vertices are of the same degree). The problem of finding $1$-factorizations in graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Deciding whether a graph has 1-factorization is usually a very difficult question. For example, it took more than 60 years and an impressive tour de force of Csaba, K\"uhn, Lo, Osthus and Treglown to prove an old conjecture of Dirac from the 1950, which says that every $d$ regular graph on $n$ vertices contains a $1$-factorization, provided that $n$ is even and $d\geq 2\lceil \frac{n}{4}\rceil-1$. In this paper we address the natural question of estimating $F(n,d)$, the number of $1$-factorizations in $d$-regular graphs on an even number of vertices, provided that $d\geq \frac{n}{2}+\varepsilon n$. Improving upon an old result of Cameron from the 1970s, we show that $F(n,d)\geq \left((1+o(1))\frac{d}{e^2}\right)^{nd/2}$, which is asymptotically best possible.

Number of 1-factorizations of regular high-degree graphs

The standard conditions in Hall's perfect matching theorem for a bipartite graph G require that all subsets from one side of G are expanding. The unbiased extension identifies mixtures of subsets f...

https://www.jstor.org/stable/10.4169/amer.math.monthly.124.1.79

Eli Shamir

Papers

Two-Sided, Unbiased Version of Hall's Marriage Theorem