Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H , each satisfying the following condition: the degree of each legal r−1-tuple contained in the complement of this side is strictly larger than n 2 . We prove that under this condition H must have a perfect matching. This answers a question of Kühn and Osthus.

In this paper, we propose an algorithm for enumerating all the perfect matchings included in a given bipartite graph G = (V,E). The algorithm is improved by the approach which we proposed at ISAAC98. Our algorithm takes O(log |V |) time per perfect matching while the current fastest algorithm takes O(|V |) time per perfect matching. Keyword: enumeration, enumerating algorithm, perfect matching.

Consider a bipartite graph G = (X; Y ;E) with real-valued weights on its edges, and suppose that G is balanced, with |X| = |Y |. The assignment problem asks for a perfect matching in G of minimum total weight. Assignment problems can be solved by linear programming, but fast algorithms have been developed that exploit their special structure. The famous Hungarian Method runs in time O(mn+n 2 lo...

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G = (V,E) with |V | = 2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function L : E → {c1, . . . , cq}, the labeled perfect matching problem consists in finding a perfect matching on G that us...

Let G be a cubic graph with a perfect matching. An edge e of G is a forcing edge if it is contained in a unique perfect matching M , and the perfect matching M is called uniquely forced. In this paper, we show that a 3-connected cubic graphwith a uniquely forced perfect matching is generated from K4 via Y → 1-operations, i.e., replacing a vertex by a triangle, and further a cubic graph with a u...