Tight upper bound on the maximum anti-forcing numbers of graphs

Let GG be a simple graph with a perfect matching. Deng and Zhang showed thatthe maximum anti-forcing number of GG is no more than the cyclomatic number.In this paper, we get a novel upper bound on the maximum anti-forcing number ofGG and investigate the extremal graphs. If GG has a perfect matching MMwhose anti-forcing number attains this upper bound, then we say GG is anextremal graph and MM is a nice perfect matching. We obtain an equivalentcondition for the nice perfect matchings of GG and establish a one-to-onecorrespondence between the nice perfect matchings and the edge-involutions ofGG, which are the automorphisms αα of order two such that vv andα(v)α(v) are adjacent for every vertex vv. We demonstrate that all extremalgraphs can be constructed from K2K2 by implementing two expansion operations,and GG is extremal if and only if one factor in a Cartesian decomposition ofGG is extremal. As examples, we have that all perfect matchings of thecomplete graph K2nK2n and the complete bipartite graph Kn,nKn,n are nice.Also we show that the hypercube QnQn, the folded hypercube FQnFQn (n≥4n≥4)and the enhanced hypercube Qn,kQn,k (0≤k≤n−40≤k≤n−4) have exactly nn,n+1n+1 and n+1n+1 nice perfect matchings respectively.