Saturating stable matchings

I relate bipartite graph matchings to stable matchings. I prove a necessary and sufficient condition for the existence of a saturating stable matching, where every agent on one side is matched, for all possible preferences. I extend my analysis to perfect stable matchings, where every agent on both sides is matched.

💡 Research Summary

The paper investigates the existence of “saturating stable matchings” in bipartite graphs, i.e., matchings that are both stable (in the Gale‑Shapley sense) and X‑saturating (every vertex on one side, X, is matched). While Hall’s theorem gives a necessary and sufficient condition for the existence of an X‑saturating matching without regard to stability, the addition of stability requires extra structural constraints.

The author defines two simple graph‑theoretic conditions for each x∈X:

1. |N(N(x))| ≤ |N(x)|, where N(x) is the set of neighbours of x and N(N(x)) the set of neighbours of those neighbours. This condition ensures that if x were unmatched, all its neighbours would already be matched to vertices within N(N(x)), which is impossible by the pigeon‑hole principle when the inequality holds.
2. There exists a neighbour y∈N(x) with degree 1. If x were unmatched, y would also be unmatched, forming a blocking pair and violating stability.

Lemma 1 shows that if either condition holds for a given x, then x must be matched in every stable matching for every possible preference profile. Lemma 2 proves the converse: if both conditions fail for some x, one can construct a preference profile that forces x to remain unmatched in every stable matching. The construction makes all neighbours of x prefer any other acceptable partner to x, and those other partners prefer neighbours of x over any alternative, guaranteeing a blocking situation for any attempted match involving x.

Theorem 1 combines these lemmas to give a clean necessary and sufficient condition: All stable matchings are X‑saturating for all preference instances if and only if every x∈X satisfies condition (1) or condition (2) (or both). The proof uses the McVitie‑Wilson theorem, which states that if a participant is unmatched in one stable marriage, they are unmatched in all; thus the existence of a single X‑saturating stable matching implies that every stable matching is X‑saturating. Lemma 3 and Corollary 1 further clarify that checking any one stable matching suffices.

The paper then extends the analysis to perfect matchings (both sides saturated). In a complete bipartite graph K_{n,n} (the classic Gale‑Shapley setting with complete preferences), condition (1) holds trivially for every vertex, so Theorem 1 guarantees that every stable matching is perfect. Theorem 2 proves that for a connected bipartite graph, the property “all stable matchings are perfect for all preference profiles” holds iff the graph is complete bipartite. The proof proceeds by induction on the number of vertices, using the same neighborhood‑size argument. For disconnected graphs, Corollary 2 shows that each connected component must be a biclique (a complete bipartite subgraph); only then does the perfect‑matching property extend to the whole graph.

Finally, the author addresses matching markets with compatibility constraints. Vertices in X and Y are partitioned (possibly with overlap) into n classes A_i and B_i. A match is acceptable only if the two vertices belong to the same class. Under “class‑wise complete” (CW‑complete) preferences—where every vertex finds every vertex of its own class acceptable—the paper proves Theorem 3: All stable matchings are X‑saturating for all preference instances iff |B_i| ≥ |A_i| for every class i. The proof again relies on condition (1): for any x, the size of its second‑order neighbourhood equals the sum of the sizes of the classes it belongs to, while its first‑order neighbourhood size equals the sum of the corresponding B_i sizes. The inequality guarantees condition (1) for all x, and Lemma 1 then yields the result. The converse follows by constructing a preference profile that leaves some x unmatched when a class violates the size condition.

Overall, the paper provides a concise, graph‑based characterization of when a bipartite market can guarantee both stability and full coverage of one side (or both sides) regardless of agents’ preferences. The conditions are easy to verify from the underlying graph structure, making them valuable for market designers who need to ensure that every participant on a given side can be matched without sacrificing stability. The extensions to perfect matchings and to markets with class‑based compatibility broaden the applicability to many real‑world allocation problems such as school admissions, residency placements, and other two‑sided matching platforms.