The graph of perfect matching polytope and an extreme problem

For graph GG, its perfect matching polytope Poly(G)Poly(G) is the convex hull of incidence vectors of perfect matchings of GG. The graph corresponding to the skeleton of Poly(G)Poly(G) is called the perfect matching graph of GG, and denoted by PM(G)PM(G). It is known that PM(G)PM(G) is either a hypercube or hamilton connected [D.J. Naddef, W.R. Pulleyblank, Hamiltonicity and combinatorial polyhedra, J. Combin. Theory Ser. B 31 (1981) 297–312; D.J. Naddef, W.R. Pulleyblank, Hamiltonicity in (0-1)-polytope, J. Combin. Theory Ser. B 37 (1984) 41–52]. In this paper, we give a sharp upper bound of the number of lines for the graphs GG whose PM(G)PM(G) is bipartite in terms of sizes of elementary components of GG and the order of GG, respectively. Moreover, the corresponding extremal graphs are constructed.