Some tight bounds on the minimum and maximum forcing numbers of graphs

Let be a simple graph with vertices and a perfect matching. We denote by and the minimum and maximum forcing numbers of , respectively. Hetyei obtained that the number of edges of graphs with a unique perfect matching is at most . Since a graph has a unique perfect matching if and only if , along this line, we generalize easily the classical result to all graphs with for , and get a non-trivial lower bound of in terms of the order and size. For bipartite graphs, we gain the corresponding stronger results. Such lower bounds enable one to obtain the minimum forcing number of some dense graphs. Further, we obtain a new upper bound of . For bipartite graphs , Che and Chen (2013) obtained that if and only if is complete bipartite graph . We completely characterize all bipartite graphs with .