Two-regular subgraphs of odd-uniform hypergraphs

TL;DR

This paper determines the maximum number of edges in large odd-uniform hypergraphs without 2-regular subgraphs, confirming a conjecture and characterizing extremal structures.

Contribution

It proves a conjecture by Mubayi and Verstra"{e}te} about the maximum edges in such hypergraphs and characterizes the extremal configurations.

Findings

Maximum edges in hypergraphs without 2-regular subgraphs identified

Extremal hypergraphs are full k-stars with a maximal matching

Conjecture of Mubayi and Verstra"{e}te verified

Abstract

Let $k\ge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $\binom{n-1}{k-1} + \lfloor\frac{n-1}{k} \rfloor$, and the equality holds if and only if $H$ is a full $k$-star with center $v$ together with a maximal matching omitting $v$. This verifies a conjecture of Mubayi and Verstra\"{e}te.