Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

Perfect matchings and the octahedron recurrence

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec Diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.     

Higher Dimensional Aztec Diamonds and a (2d + 2)-Vertex Model

Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the square-ice model, we consider a higher dimensional analog of this phenomenon. For d 1, we construct d-uniform hypergraphs which generalize the Aztec diamonds and we consider a companion d-dimensional statistical model (called the 2d + 2-vertex model) whose free energy is given by the logarithm of the number of perfect matchings of our hypergraphs. We prove that the limit defining the free energy per site of the 2d + 2-vertex model exists and we obtain bounds for it. As a consequence, we obtain an especially good asymptotical approximation for the number of matchings of our hypergraphs.     

Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid G _n of order n is similar to the disjoint union of two copies of the quartered Aztec diamond QAD _n−1 of order n−1 with the path P _n ⁽²⁾ on n vertices having edge weights equal to 2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. It follows in particular that the characteristic polynomials of the above graphs satisfy the equality P(G _n )=P(P _n ⁽²⁾)[P(QAD _n−1)]². On the one hand, this provides a combinatorial explanation for the “squarishness” of the characteristic polynomial of the square grid—i.e., that it is a perfect square, up to a factor of relatively small degree. On the other hand, as formulas for the characteristic polynomials of the path and the square grid are well known, our equality determines the characteristic polynomial of the quartered Aztec diamond. In turn, the latter allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described “linear squarishness” property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases—graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases. Research supported in part by NSF grant DMS-0500616.     

Perfect path double covers of graphs

A perfect path double cover (PPDC) of a graph G on n vertices is a family ?? of n paths of G such that each edge of G belongs to exactly two members of ?? and each vertex of G occurs exactly twice as an end of a path of ??. We propose and study the conjecture that every simple graph admits a PPDC. Among other things, we prove that every simple 3-regular graph admits a PPDC consisting of paths of length three.     

Computing a pyramid partition generating function with dimer shuffling

We verify a recent conjecture of Kenyon/Szendr?i by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁x₂−x₃x₄=0}⊂C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp [Noam Elkies, Greg Kuperberg, Michael Larsen, James Propp, Alternating sign matrices and domino tilings. II, J. Algebraic Combin. 1 (3) (1992) 219-234].     

On Perfect Matching Coverings and Even Subgraph Coverings

A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G). A cubic graph G is called a Kotzig graph if G has a 3‐edge‐coloring such that each pair of colors forms a hamiltonian circuit (introduced by R. Häggkvist, K. Markström, J Combin Theory Ser B 96 (2006), 183–206). In this article, we prove that if there is a vertex w of a cubic graph G such that , the graph obtained from by suppressing all degree two vertices is a Kotzig graph, then G has a Berge covering. We also obtain some results concerning the so‐called 5‐even subgraph double cover conjecture.     

Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

A Toeplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Johansson.     

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.     

On the Choosability of Claw-Free Perfect Graphs

It has been conjectured that for every claw-free graph G the choice number of G is equal to its chromatic number. We focus on the special case of this conjecture where G is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most 4.     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

Enumerating Perfect Matchings in n-Cubes

The perfect matchings in the n-cube have earlier been enumerated for n?≤?6. A dynamic programming approach is here used to obtain the total number of perfect matchings in the 7-cube, which is 391 689 748 492 473 664 721 077 609 089. The number of equivalence classes of perfect matchings is further shown to be 336 in the 5-cube, 356 788 059 in the 6-cube and 607 158 046 495 120 886 820 621 in the 7-cube. The techniques used can be generalized to arbitrary bipartite and general graphs.     

Perfect matchings in regular bipartite graphs

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk – 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank – 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr 0, stillG – S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.     

On the Intersection Number of Matchings and Minimum Weight Perfect Matchings of Multicolored Point Sets

Let P and Q be disjoint point sets with 2r and 2s elements respectively, and M₁ and M₂ be their minimum weight perfect matchings (with respect to edge lengths). We prove that the edges of M₁ and M₂ intersect at most |M₁|+|M₂|−1 times. This bound is tight. We also prove that P and Q have perfect matchings (not necessarily of minimum weight) such that their edges intersect at most min{r,s} times. This bound is also sharp. Supported by PAPIIT(UNAM) of México, Proyecto IN110802 Supported by FAI-UASLP and by CONACYT of México, Proyecto 32168-E Supported by CONACYT of México, Proyecto 37540-A     

Perfect matchings extending on subcubes to Hamiltonian cycles of hypercubes

Recently, Fink [J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, J. Combin. Theory Ser. B 97 (2007) 1074-1076] affirmatively answered Kreweras’ conjecture asserting that every perfect matching of the hypercube extends to a Hamiltonian cycle. We strengthen this result in the following way. Given a partition of the hypercube into subcubes of nonzero dimensions, we show for every perfect matching of the hypercube that it extends on these subcubes to a Hamiltonian cycle if and only if it interconnects them.     

Exponentially many perfect matchings in cubic graphs

We show that every cubic bridgeless graph G has at least ²|V(G)|/3656 perfect matchings. This confirms an old conjecture of Lovász and Plummer.     

Generating Function of the Tilings of an Aztec Rectangle with Holes

We consider a generating function of the domino tilings of an Aztec rectangle with several unit squares removed from the boundary. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.     

Perfect matchings in uniform hypergraphs with large minimum degree

A perfect matching in a k-uniform hypergraph on n vertices, n divisible by k, is a set of n/k disjoint edges. In this paper we give a sufficient condition for the existence of a perfect matching in terms of a variant of the minimum degree. We prove that for every k≥3 and sufficiently large n, a perfect matching exists in every n-vertex k-uniform hypergraph in which each set of k−1 vertices is contained in n/2+Ω(logn) edges. Owing to a construction in [D. Kühn, D. Osthus, Matchings in hypergraphs of large minimum degree, J. Graph Theory 51 (1) (2006) 269–280], this is nearly optimal. For almost perfect and fractional perfect matchings we show that analogous thresholds are close to n/k rather than n/2.     

Sparsely intersecting perfect matchings in cubic graphs

In 1971, Fulkerson made a conjecture that every bridgeless cubic graph contains a family of six perfect matchings such that each edge belongs to exactly two of them; equivalently, such that no three of the matchings have an edge in common. In 1994, Fan and Raspaud proposed a weaker conjecture which requires only three perfect matchings with no edge in common. In this paper we discuss these and other related conjectures and make a step towards Fulkerson’s conjecture by proving the following result: Every bridgeless cubic graph which has a 2-factor with at most two odd circuits contains three perfect matchings with no edge in common.     

A fractional version of the Erdős-Faber-Lovász conjecture

LetH be any hypergraph in which any two edges have at most one vertex in common. We prove that one can assign non-negative real weights to the matchings ofH summing to at most |V(H)|, such that for every edge the sum of the weights of the matchings containing it is at least 1. This is a fractional form of the Erds-Faber-Lovász conjecture, which in effect asserts that such weights exist and can be chosen 0,1-valued. We also prove a similar fractional version of a conjecture of Larman, and a common generalization of the two.Supported in part by NSF grant MCS 83-01867, AFOSR Grant 0271 and a Sloan Research Fellowship