Let $G=(L\cup R,E)$ be a bipartite graph with $n$ nodes on each side. Find the upper bound on $n$ such that if any 2 vertices in $L$ have at-least 2 neighbors in $R$, Then $G$ has a perfect matching.
2026-03-28 16:58:14.1774717094
Showing a bipartite graph has a perfect matching with given number of neighbors
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This is false for $n\geq 5$. Pick two vertices $l,l'$ in $L$ and two vertices $r,r'$ in $R$. Consider the bipartite graph in which the only edges which exist are those which have at least one of $l,l',r$ or $r'$ as a vertex.
There is no perfect matching as at most $4$ edges can belong to the matching.
Example for $n=5$: