In a graph, perfect matching cut is an edge that matching. (pmc) the problem of deciding whether given graph has cut, and known to be NP-complete. We revisit show pmc remains NP-complete when restricted bipartite graphs maximum degree 3 arbitrarily large girth. Complementing this hardness result, we give two classes in which polynomial-time solvable. The first one includes claw-free without ind...

Let G = (V,E) be an edge-colored graph, i.e., G is assigned a surjective function C : E → {1, 2, · · · , r}, the set of colors. A matching of G is called heterochromatic if its any two edges have different colors. Let (B,C) be an edge-colored bipartite graph and d(v) be color degree of a vertex v. We show that if d(v) ≥ k for every vertex v of B, then B has a heterochromatic matching of cardina...

For two bipartite graphs G and G , a bijection ψ : E(G) → E(G) is called a (perfect) matching preserver provided that M is a perfect matching in G if and only if ψ(M) is a perfect matching in G. We characterize bipartite graphs G and G which are related by a matching preserver and the matching preservers between them.

It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analogue of this result for hypergraphs. We also prove several related results which guarantee the existence of almost perfect matchings in r-uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best poss...

Let G = (V, E) be an (edge-)colored graph, i.e., G is assigned a mapping C : E → {1, 2, · · · , r}, the set of colors. A matching of G is called heterochromatic if its any two edges have different colors. Unlike uncolored matchings for which the maximum matching problem is solvable in polynomial time, the maximum heterochromatic matching problem is NP-complete. This means that to find both suff...

Traffic analysis is the best known approach to uncover relationships amongst users of anonymous communication systems, such as mix networks. Surprisingly, all previously published techniques require very specific user behavior to break the anonymity provided by mixes. At the same time, it is also well known that none of the considered user models reflects realistic behavior which casts some dou...

For a certain graph obtained by adding extra vertices and edges to the triangular lattice graph, Propp conjectured that the number of perfect matchings of such a graph is always divisible by 3. In this note we prove this conjecture. In a graph G, a matching is a set of edges such that no two edges are incident to each other. A matching in a graph is called perfect if every vertex is incident wi...

It is a longstanding open problem whether there exists a polynomial size description of the perfect matching polytope. We give a partial answer to this question by proving the following result. The polyhedron defined by the constraints of the perfect matching polytope which are active at a given perfect matching can be obtained as the projection of a compact polyhedron. Thus there exists a comp...

Let G be a graph with even order. Let M be a matching in G and x1, x2, . . . , x2r , be the M-unsaturated vertices in G. Then G has a perfect matching if and only if there are r independentM-augmenting paths joining the 2r vertices in pairs. LetG be a graph with a perfect matchingM . It is proved that G is 2k-critical if and only if for any 2k vertices u1, u2, . . . , u2k in G, there are k inde...