On the perfect 1-factorisation problem for circulant graphs of degree 4

A 1-factorisation of a graph G is a partition of the edge set of G into 1 factors (perfect matchings); a perfect 1-factorisation of G is a 1-factorisation of G in which the union of any two of the 1-factors is a Hamilton cycle in G. It is known that for bipartite 4-regular circulant graphs, having order 2 (mod 4) is a necessary (but not sufficient) condition for the existence of a perfect 1-factorisation. The only known non-bipartite 4-regular circulant graphs that admit a perfect 1-factorisation are trivial (on 6 vertices). We prove several construction results for perfect 1-factorisations of a large class of bipartite 4-regular circulant graphs. In addition, we show that no member of an infinite family of non-bipartite 4-regular circulant graphs admits a perfect 1-factorisation. This supports the conjecture that there are no perfect 1-factorisations of any connected non-bipartite 4-regular circulant graphs of order at least 8

Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs

We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain [JS89] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least $\Omega(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.Comment: 14 pages, no figure

The Cost of Perfection for Matchings in Graphs

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs

Rainbow perfect matchings in r-partite graph structures

A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft

Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced