The Kőnig graph process

Say that a graph G has property K if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set N := ( n 2 ) and let e1, e2, . . . eN be a uniformly random ordering of the edges of Kn, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm+1 is obtained from Gm by adding the edge em+1 exactly if Gm ∪ {em+1} has property K. We analyse the behaviour of this process, focusing mainly on two questions: What can be said about the structure of GN and for which m will Gm contain a perfect matching?

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