On a Generalization of the Ryser‐Brualdi‐Stein Conjecture

A rainbow matching for (not necessarily distinct) sets of hypergraph edges is a matching consisting of k edges, one from each . The aim of the article is twofold—to put order in the multitude of conjectures that relate to this concept (some first presented here), and to prove partial results on one of the central conjectures.     

Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

Small cycle double covers of products I: Lexicographic product with paths and cycles

Bondy conjectured that every simple bridgeless graph has a small cycle double cover (SCDC). We show that this is the case for the lexicographic products of certain graphs and along the way for the Cartesian product as well. Specifically, if G does not have an isolated vertex then G □ P₂ and G □ C_2k have SCDCs. If G has an SCDC then so does G □ P_k, k > 2 and G □ C_{2k + 1}. We use these Cartesian results to show that P_2j[G] (j ≥ 1) and C_k[G] (k ≠ 3, 5, 7) have SCDCs. Also, if G has an SCDC then so does P_{2j + 1}[G] (j ≥ 4). The results for the lexicographic product are harder and, in addition to the Cartesian results, require certain decompositions of K_n,n into perfect matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 99–123, 2008     

Matchings in hypergraphs of large minimum degree

It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analog of this result for hypergraphs. We also prove several related results that guarantee the existence of almost perfect matchings in r‐uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best possible. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 269–280, 2006     

Monotonicity and Consistency in Matching Markets

Objective: To obtain axiomatic characterizations of the core of one-to-one and one-to-many matching markets. Methods: The axioms recently applied to characterize the core of assignment games were adapted to the models of this paper. Results: The core of one-to-one matching markets is characterized by two different lists of axioms. The first one consists of weak unanimity, population monotonicity, and Maskin monotonicity. The second consists of weak unanimity, population monotonicity, and consistency. If we allow for weak preferences, the core is characterized by weak unanimity, population monotonicity, Maskin monotonicity, and consistency. For one-to-many matchings, the same lists as for the case of strict preferences characterize the core. Conclusions: The cores of the discrete matching markets are characterized by axioms that almost overlap with the axioms characterizing the core of the continuous matching markets. This provides an axiomatic explanation for the observations in the literature that almost parallel properties are obtained for the core of the two models. We observe that Maskin monotonicity is closely related to consistency in matching marketsThis research is financially supported by Waseda University Grant for Special Research Projects #2000A−887, 21COE-GLOPE, and Grant-in-Aid for Scientific Research #15530125, JSPS. This paper was presented at the 7th. International Meeting of the Society for Social Choice and Welfare held in Osaka, Japan. The comments of the participants are gratefully acknowledged. The author thanks Professors William Thomson, Eiichi Miyagawa and anonymous referees for their valuable comments and suggestions. Any remaining errors are independent     

Total dual integrality and b-matchings

Schrijver has shown that any rational polyhedron is the solution set of a unique minimal integer TDI linear system. We characterize this system for the case of b-matchings, one of the few known cases for which such a system is strictly larger than a minimal linear system sufficient to define the polyhedron.     

Fulkersonʼs Conjecture and Loupekine snarks

Fulkersonʼs Conjecture says that every bridgeless cubic graph has six perfect matchings such that each edge belongs to exactly two of them. In 1976, F. Loupekine created a method for constructing new snarks from already known ones. We consider an infinite family of snarks built with Loupekineʼs method, and verify Fulkersonʼs Conjecture for this family.     

K best solutions to combinatorial optimization problems

We review the Lawler-Murty [24,20] procedure for finding theK best solutions to combinatorial optimization problems. Then we introduce an alternative algorithm which is based on a binary search tree procedure. We apply both algorithms to the problems of finding theK best bases in a matroid, perfect matchings, and best cuts in a network.Partially supported by the National Science Foundation, No. ECS-8412926.     

Singular values of Gaussian matrices and permanent estimators

We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad‐connectedness condition. Using these estimates and concentration of measure for the spectrum of Gaussian matrices with independent entries, we prove that for a large class of graphs satisfying an appropriate expansion property, the Barvinok–Godsil‐Gutman estimator for the permanent achieves sub‐exponential errors with high probability. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 183–212, 2016     

Matching signatures and Pfaffian graphs

Perfect matchings of k-Pfaffian graphs may be enumerated in polynomial time on the number of vertices, for fixed k. In general, this enumeration problem is #P-complete. We give a Composition Theorem of 2r-Pfaffian graphs from r Pfaffian spanning subgraphs. Constructions of k-Pfaffian graphs known prior to this seem to be of a very different and essentially topological nature. We apply our Composition Theorem to produce a bipartite graph on 10 vertices that is 6-Pfaffian but not 4-Pfaffian. This is a counter-example to a conjecture of Norine (2009) [8], which states that the Pfaffian number of a graph is a power of four.