The graph of perfect matching polytope and an extreme problem

For graph G , its perfect matching polytope P o l y ( G ) is the convex hull of incidence vectors of perfect matchings of G . The graph corresponding to the skeleton of P o l y ( G ) is called the perfect matching graph of G , and denoted by P M ( G ) . It is known that P M ( 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 G whose P M ( G ) is bipartite in terms of sizes of elementary components of G and the order of G , respectively. Moreover, the corresponding extremal graphs are constructed.