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# sol8 - MATH 239 Fall 2008 Assignment 8 Solutions Total 20...

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MATH 239 Fall 2008 Assignment 8 Solutions Total 20 marks 1. The purpose of this problem is to prove that every non-maximum matching admits an aug- menting path. Let M be a matching in a graph G . Let N be a matching which is larger than M . Let H be the subgraph of G whose edges are ( M N ) \ ( M N ) and whose vertices are the set of vertices of G . (a) Show that every component of H consists of either a path or a cycle. (b) Prove that every path in H is an alternating path of G with respect to M . (c) Prove that at least one path in H is an augmenting path of G with respect to M . Solution. (a) (2 marks) Every vertex of H is incident to at most two edges of M N , since M and N are matchings. Therefore, every vertex of H has degree equal to either zero, one or two. It follows that every component of H is a path or a cycle (where we consider a vertex of degree zero to be a path of length zero). (b) (2 marks) Every vertex of H of degree two is incident with an edge from each of M and N . Hence every path in H is an alternating path of G with respect to M , and an alternating path with respect to N . (c) (2 marks) Observe that ( M N ) \ ( M N ) is equal to the set of edges which belong to exactly one of M or N . Since N has more edges than M , the subgraph H has more edges of N than of M .

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