Disjoint Y-Y paths covering X under the double Hall property

Prove that every dHp bigraph G=(X,Y) satisfying |N(X)|≥|X|+1 contains a collection of disjoint Y-Y paths whose union covers all vertices of X.

Background

This conjecture is introduced as a weaker companion to the high-degree cycle-cover conjecture. Instead of requiring one cycle through all vertices of X, it asks for disjoint paths with endpoints in Y whose union covers X.

The paper proves that this conjecture and the high-degree cycle-cover conjecture are equivalent, but does not establish either one in full generality. It verifies the relevant bounded-size cases needed to obtain the result for k≤7.

References

If a dHp bigraph $G=(X, Y)$ satisfies $|N(X)| \geq |X| + 1$, then there exists a collection of disjoint $Y-Y$ paths whose union covers all of $X$.

— Bipartite graphs with the double Hall property  (2502.10903 - Chen et al., 15 Feb 2025) in Conjecture 4, Introduction