Cycle cover under a high-degree condition on Y

Prove that for every nonnegative integer k, every dHp bigraph G=(X,Y) with |X|=n>max{2k+1,k(k+1)} and deg(y)≥n−k for every y∈Y contains a cycle covering all vertices of X.

Background

This conjecture is a degree-restricted weakening of Salia’s conjecture. It imposes a lower bound on every vertex degree in the Y-part while retaining the double Hall property.

The paper proves the conjecture for k≤7, using its result for snp bigraphs with at most seven vertices in X. The statement for general nonnegative k remains unresolved.

References

Let $k$ be a nonnegative integer, and consider a dHp bigraph $G=(X, Y)$. If $|X| = n > \max {2k+1, k(k+1)}$ and $\deg(y) \ge n - k$ for all $y \in Y$, then there is a cycle in $G$ covering all vertices of $X$.

— Bipartite graphs with the double Hall property  (2502.10903 - Chen et al., 15 Feb 2025) in Conjecture 3, Introduction