Matching graphs of Hypercubes and Complete Bipartite Graphs

Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC perfect matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution. In this article, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting perfect matchings in a crucial way. Our main new idea is an NC algorithm for finding a face of the perfect matching polytope at which a set (which could be polynomially large) of conditions, involving constraints of the polytope, are simultaneously satisfied. Several other ideas are also needed, such as finding, in NC, a point in the interior of the minimum-weight face of this polytope and finding a balanced tight odd set.

On the mutually independent Hamiltonian cycles in faulty hypercubes

Almost color-balanced perfect matchings in color-balanced complete graphs

Let M be a perfect matching of a graph G. The smallest number of edges whose removal to make M as the unique perfect matching in the resulting subgraph is called the anti-forcing number of M. The anti-forcing spectrum of G is the set of anti-forcing numbers of all perfect matchings in G, denoted by $$\hbox {Spec}_{af}(G)$$Specaf(G). In this paper, we show that any finite set of positive integers can be the anti-forcing spectrum of a graph. We present two classes of hexagonal systems whose anti-forcing spectra are integer intervals. Finally, we show that determining the anti-forcing number of a perfect matching of a bipartite graph with maximum degree four is a NP-complete problem.

Let W1,…,Wn be independent random subsets of [m]={1,…,m}. Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m,d) on the vertex set {W1,…Wn}, defined by the adjacency relation: Wi∼Wj whenever |Wi∩Wj|≧s. For even n we show that as n,m→∞ the edge density threshold for the property that Gs(n,m,d) contains a perfect matching is asymptotically the same as that for Gs(n,m,d) being connected.

on vertex set , , with density d>2e and all vertex degrees not too far from d, has about as many perfect matchings as a corresponding random bipartite graph, i.e. about . In this paper we utilize that result to prove that with probability quickly approaching one, a perfect matching drawn randomly from G is spread evenly, in the sense that for any large subsets of vertices and , the number of edges of the matching spanned between S and T is close to |S||T|/n (c.f. Lemma 1). As an application we give an alternative proof of the Blow-up Lemma of Komlos, Sarkozy and Szemeredi [10].

Trees possessing no Kekul ́e structures (i.e., perfect matching) with the minimal Estrada index are considered. Let T_n be the set of the trees having no perfect matchings with n vertices. When n is odd and n ≥ 5, the trees with the smallest and the second smallest Estrada indices among T_n are obtained. When n is even and n ≥ 6, the tree with the smallest Estrada index in T_n is deduced.

Given two k-graphs H and F, a perfect F-packing in H is a collection of vertex-disjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F-packing is simply a perfect matching. For a given fixed F, it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F-packing is NP-complete. Indeed, if k≥3, the corresponding problem for perfect matchings is NP-complete [17,7] whilst if k=2 the problem is NP-complete in the case when F has a component consisting of at least 3 vertices [14].In this paper we give a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect F-packings is polynomial time solvable. We then give three applications of this tool: (i) Given 1≤ℓ≤k−1, we give a minimum ℓ-degree condition for which it is polynomial time solvable to determine whether a k-graph satisfying this condition has a perfect matching; (ii) Given any graph F we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect F-packing; (iii) We also prove a similar result for perfect K-packings in k-graphs where K is a k-partite k-graph.For a range of values of ℓ,k (i) resolves a conjecture of Keevash, Knox and Mycroft [20]; (ii) answers a question of Yuster [47] in the negative; whilst (iii) generalises a result of Keevash, Knox and Mycroft [20]. In many cases our results are best possible in the sense that lowering the minimum degree condition means that the corresponding decision problem becomes NP-complete.

A subgraph H of a graph G with perfect matching is nice if G−V(H) has perfect matching. It is well-known that all fullerene graphs have perfect matchings and that all fullerene graphs contain some small connected graphs as nice subgraphs. In this contribution, we consider fullerene graphs arising from smaller fullerenes via the leapfrog transformation, and show that in such graphs, each pair of (necessarily disjoint) pentagons is nice. That answers in affirmative a question posed in a recent paper on nice pairs of odd cycles in fullerene graphs.

Our topic is an extension of the following classical result of Hall to hypergraphs: A bipartite graph G contains a perfect matching if and only if for each independent set X of vertices, at least |X| vertices of G are adjacent to some vertex of X. Berge generalized the concept of bipartite graphs to hypergraphs by defining a hypergraph G to be balanced if each odd cycle in G has an edge containing at least three vertices of the cycle. Based on this concept, Conforti, Cornuejols, Kapoor, and Vuskovic extended Hall's result by proving that a balanced hypergraph G contains a perfect matching if and only if for any disjoint sets A and B of vertices with |A| > |B|, there is an edge in G containing more vertices in A than in B (for graphs, the latter condition is equivalent to the latter one in Hall's result). Their proof is non-combinatorial and highly based on the theory of linear programming. In the present paper, we give an elementary combinatorial proof.

The perfect matching graph of a graph G, denoted by M(G), has one vertex for each perfect matching of G and two matchings are adjacent if their symmetric difference is a cycle of G. Let C be a family of cycles of G. The perfect matching graph defined byC is the spanning subgraph M(G, C) of M(G) in which two perfect matchings L and N are adjacent only if \(L \varDelta N\) lies in C. We give a necessary condition and a sufficient condition for M(G, C) to be connected. We also give examples of graphs and of families of cycles for which the sufficient condition is satisfied.

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.
 In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching $M$ from a cubic graph $G$ leaves a disjoint union of cycles. Contracting these cycles yields a new graph $G_M$. The graph $G$ is star-like if $G_M$ is a star for some perfect matching $M$, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

The problem of determining whether a graph has a Hamilton cycle is NP-complete whereas there exists a polynomial algorithm to determine whether a graph has a perfect 2-matching. These two problems are related to the question of determining whether a graph has a perfect triangle-free 2-matching. We give a polynomial algorithm to answer this question and to find a perfect triangle-free 2-matching if one exists.

Spectral radius and matchings in graphs

Independent sets versus perfect matchings