We establish cutoff for a natural random walk (RW) on the set of perfect matchings (PMs), based “rewiring”. An n-PM is pairing 2n objects. The k-PM RW selects k pairs uniformly at random, disassociates corresponding 2k objects, then chooses new these objects random. equilibrium distribution uniform over all n-PMs. 2-PM was first introduced by Diaconis and Holmes (Proc. Natl. Acad. Sci. USA 95 (...

Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$, and let $F_i \subseteq{[n]\choose 4}$, where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose 3}-{3n/4\choose 3}$ edges, then $\{F_1, \ldots ,F_{n/4}\}$ admits rainbow matching, i.e., set $n/4$ edges consisting one edge from $F_i$. This generalizes deep result Khan J. Combin. Theory Ser. B,...

The semirandom graph process is a single player game in which the initially presented an empty on $n$ vertices. In each round, vertex $u$ to independently and uniformly at random. then adaptively selects $v$ adds edge $uv$ graph. For fixed monotone property, objective of force satisfy this property with high probability as few rounds possible. We focus problem constructing perfect matching part...

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem that was introduced by Park and Ihm. The burnt pancake graph is a more complex variant of the pancake graph. In this talk, we examine the properties ...

Let G be a bipartite graph with bicoloration {A, B}, |A| = |B|, and let w : E(G) -» K where K is a finite abelian group with k elements. For a subset S c E(G) let w(S) = IIeE s (e).A Perfect matching M c E(G) is a w-matching if w(M) = 1. A characterization is given for all w's for which every perfect matching is a w-matching. It is shown that if G = Kk+1,k+1 then either G has no w-matchings or ...

A graph H is single-crossing if it can be drawn in the plane with at most one crossing. For any single-crossing graph H, we give an O(n4) time algorithm for counting perfect matchings in graphs excluding H as a minor. The runtime can be lowered to O(n1.5) when G excludes K5 or K3,3 as a minor. This is the first generalization of an algorithm for counting perfect matchings in K3,3free graphs (Li...

The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent if their symmetric difference forms the edge set of a hexagon of B . A family P of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if B−P contains at least one perfect matching, or if B − P is empty. It is proven that there exists a surjective map ...