Dirac's theorem for linear hypergraphs
Seonghyuk Im, Hyunwoo Lee
TL;DR
This work extends Dirac's theorem to linear $k$-uniform hypergraphs by proving that a minimum degree of at least $\frac{n}{k} + C$ guarantees a matching covering at least $(1-\varepsilon)n$ vertices, with the constant $C$ depending on $\varepsilon$ and $k$; and shows that a weaker threshold of $\delta(H) \ge (\frac{1}{k}+o(1))n$ yields almost spanning linear cycles, almost spanning hypertrees with $o(n)$ leaves, and long subdivisions of $o(\sqrt{n})$-vertex graphs. The approach builds a regularization step to obtain a pseudorandom fractional matching and then uses the Pippenger–Spencer theorem to realize near-tilings by hypertrees, augmented by a reservoir method to embed almost-spanning subdivisions. The results are asymptotically tight and illuminate the limits of perfect matchings under degree constraints in linear hypergraphs, advancing the understanding of spanning near-structures in sparse hypergraphs. The work also develops a toolbox of hypertree structure lemmas and connection techniques that may apply to broader embedding problems in hypergraphs.
Abstract
Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $δ(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex $k$-uniform linear hypergraph $H$ with minimum degree at least $\frac{n}{k} + Ω(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This minimum degree condition is asymptotically tight and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if $δ(H) \geq (\frac{1}{k}+o(1))n$, then $H$ contains almost spanning linear cycles, almost spanning hypertrees with $o(n)$ leaves, and ``long subdivisions'' of any $o(\sqrt{n})$-vertex graphs.