So I'm trying to solve a problem in graph theory. I painted a couple of trees with perfect matchings and a couple of graphs without perfect matchings. And I have realized that when I paint non perfect matchings #Vertices $|V| = 2k+1$ and # edges $E(G) = 2k$ and for perfect matchings $|V| = 2k$ and $E(G) = 2k+1$. Is that always the case in any tree?
2026-03-26 22:55:45.1774565745
Perfect matchings and parity of the vertices and the edges in a tree
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If a graph has a perfect matching, then indeed the number of vertices must be even, because every vertex must be saturated by an edge of the matching, and every edge of the matching saturates exactly two vertices. In the case of a tree we have $|E| = |V| + 1$, so the last statement is true.
However the other way around is false. Suppose we have the graph ({1,2,3,4}, {12,13,14}), this is a star with $3$ leafs. This graph has an even number of vertices, but has no perfect matching.