Let G = (V,E) be a graph and let M1,M2 ⊂ E be two maximum matchings in G. Show that any path in G whose edges alternate between M1 and M2 and is maximal with respect to this property must contain equal number of edges from M1 and M2.
2026-03-27 07:49:53.1774597793
can two different maximum matchings M1 and M2 for a graph G = (V,E) share some of the same matchings?
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Let P:v1-v2-...-vn be a path in G whose edges alternate between matchings M1 and M2 and P cannot be extended and still have that property. If P has an even number of edges then it does have just as many edges in M1 as in M2. We can show that P does not have an odd number of edges as follows: Suppose P starts and ends with edges in M1. The edges of M2 that are NOT on P must be disjoint from P, that is, they can't have a vertex that lies on P. Let M3 be the matching made by taking the union of the set of edges of M1 lying on P and the set of edges in M2 NOT lying on P. Note that M3 is a larger matching than M1 or M2 contradicting the fact that M1 and M2 are maximum.