A bipartite graph consists of two disjoint vertex sets, where vertices of one set can only be joined with an edge to vertices in the opposite set. Hall's theorem gives a necessary and sufficient condition for a bipartite graph to have a saturating matching , meaning every vertex in one set is matchedto some vertex in the other in a one-to-one correspondence. When we imagine vertices as agents and let them have preferences over other vertices, we have the classic stable marriage problem introduced by Gale and Shapley, who showed that one can always find a matching that is stable with respect to agent's preferences. These two results often clash: saturating matchings are not always stable, and stable matchings are not always saturating. I prove a simple necessary and sufficient condition for every stable matching being saturating for one side. I show that this result subsumes and generalizes some previous theorems in the matching literature. I find a necessary and sufficient condition for stable matchingsbeing saturating on both sides , also known as perfect matchings. These results could have important implications for the analysis of numerous real-world matching markets .
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 .
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A bipartite graph consists of two disjoint vertex sets, where vertices of one set can only be joined with an edge to vertices in the opposite set. Hall's theorem gives
I relate bipartite graph matchings to stable matchings. I prove
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a bipartite graph to have a saturating matching
the existence of a saturating stable matching, where every agent on one side is matched
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meaning every vertex in one set is matchedto some vertex in the other in a one-to-one correspondence. When we imagine vertices as agents and let them have preferences over other vertices, we have the classic stable marriage problem introduced by Gale and Shapley, who showed that one can always find a matching that is stable with respect to agent's preferences. These two results often clash: saturating matchings are not always stable, and stable matchings are not always saturating. I prove a simple necessary and sufficient condition for every stable matching being saturating for one side. I show that this result subsumes and generalizes some previous theorems in the matching literature. I find a necessary and sufficient condition for stable matchingsbeing saturating
for all possible preferences. I extend my analysis to perfect stable matchings, where every agent
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, also known as perfect matchings. These results could have important implications for the analysis of numerous real-world matching markets
is matched