A sufficient condition for a hypergraph to have a Berge-$k$-factor

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.

💡 Research Summary

The paper investigates the existence of Berge‑k‑factors in hypergraphs by translating hypergraph properties into the language of bipartite incidence graphs. For a hypergraph H with vertex set V(H) and edge set E(H), the incidence bipartite graph I(H) has bipartition (E(H), V(H)), where an edge‑vertex pair is adjacent exactly when the hyperedge contains the vertex. This construction allows the authors to reformulate notions of connectivity, toughness, and factorisation for hypergraphs as corresponding concepts in the bipartite graph I(H).

The central notion introduced is Y‑toughness τY(G) of a bipartite graph G