Kahn--Lovász-type inequalities for graph factors

Abstract

The Kahn--Lovász theorem gives a sharp upper bound on the number of perfect matchings in a graph in terms of its degree sequence, extending the classical Brégman--Minc inequality for bipartite graphs. In this paper, we establish an asymptotically sharp extension of the Kahn--Lovász theorem to $F$-factors for every Hamiltonian graph $F$. As a consequence, we asymptotically determine the maximum number of $F$-factors in an $n$-vertex $m$-edge graph, yielding an $F$-factor analogue of Kruskal--Katona-type theorems. We also prove a multigraph analogue of the Kahn--Lovász theorem. Combining this with our results for Hamiltonian graphs, we obtain an asymptotically sharp Kruskal--Katona-type bound for a further class of connected graphs $F$, including those containing two vertex-disjoint cycles of equal length whose union spans $V(F)$.