A sufficient condition for a hypergraph to have a Berge-$k$-factor
Yuping Gao, Songling Shan, Gexin Yu
TL;DR
The paper studies Berge-$k$-factors in hypergraphs by translating a hypergraph $H$ into its incidence bipartite graph and establishing a toughness-based sufficient condition for the existence of a $$(\{0,2\},k)$$-factor in the associated bipartite graph, which corresponds to a Berge-$k$-factor in $H$. The authors develop a parity-factor framework using partial parity $(g,f)$-factors, $Y$-toughness, and barrier theory, proving a key parity condition $\delta_G(A,B) \equiv 0 \pmod{2}$ when $k|Y|$ is even. The main result, Theorem 2, states that if $\tau_{Y}(G) \ge k$, $k|Y|$ is even, and $|Y| \ge k+1$, then $G$ has a $(\{0,2\},k)$-factor; this yields that every $k$-tough hypergraph $H$ has a Berge-$k$-factor when $k|V(H)|$ is even and $|V(H)| \ge k+1$, and recovers the classical EJKS 1985 result for graphs. The approach blends adapted factor theory for bipartite graphs with new techniques for hypergraph cutsets, providing a unified incidence-based framework with sharp toughness-based criteria for Berge-$k$-factors.
Abstract
For any graph (hypergraph) $G$ with vertex set $V$ and edge set $E$, we define its incidence bipartite graph $\mathcal{I}(G)$ as the bipartite graph with bipartition $(E, V)$, where an edge $e \in E$ is adjacent to a vertex $v \in V$ in $\mathcal{I}(G)$ if and only if $e$ is incident to $v$ in $G$. This representation allows all concepts and properties of $G$ to be reformulated in terms of those of $\mathcal{I}(G)$. In this paper, we investigate the notions of graph toughness and $k$-factors in bipartite graphs through this incidence perspective. As an application, our result implies the classic theorem of Enomoto, Jackson, Katerinis, and Saito: for any integer $k \geq 1$, a $k$-tough graph $G$ has a $k$-factor if $k |V(G)|$ is even and $|V(G)| \geq k+1$. Furthermore, we extend this result to hypergraphs, without requiring uniformity.