The matching preclusion problem, introduced by Brigham et al. [Perfect-matching preclusion, Congressus Numerantium 174 (2005) 185-192], studies how to effectively make a graph have neither perfect matchings nor almost perfect matchings by deleting as small a number of edges as possible. Extending this concept, we consider a more general matching preclusion problem, called the strong matching pr...

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini– Schramm convergent sparse graph sequence. We prove that the normalized logarithm o...

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O(2d) time and exponential space for a constant εd depending only on d. Thus, we generalize the recent results of Björklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorit...

The importance of Pfaffian orientations stems from the fact that if a graph G is Pfaffian, then the number of perfect matchings of G (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is kn...

The determinant method of Kasteleyn gives a computing the number perfect matchings planar bipartite graph. In addition, results Bernardi exhibit bijection between spanning trees graph and elements its Jacobian. this paper, we explore an adaptation Bernardi’s results, providing simply transitive group action cokernel on set matchings, when in question is form $$G^+$$ , as defined by Kenyon, Prop...