Salia’s double Hall cycle-cover conjecture

Prove that every dHp bigraph G=(X,Y) with |X|≥2 contains a cycle containing all vertices of X.

Background

The double Hall property requires |N²(S)|≥|S| for every subset S⊆X with |S|≥2. It is weaker than the super-neighborhood property in the relevant range, and therefore Salia’s conjecture is a weakening of the super-neighborhood cycle-cover conjecture.

The paper establishes this conclusion for |X|≤7 and for several additional degree-restricted classes, but leaves the unrestricted conjecture unresolved. It also shows that the conjecture is closely related to weaker conjectures involving high degrees in Y and collections of disjoint Y-Y paths.