For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for snarks, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured.In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on n ≤ 36 vertices. Previously lists up to n = 28 vertices have been published.In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.

Given integers n ≥ k > l ≥ 1 and a k-graph F with |V (F )| divisible by n, define t k l (n, F ) to be the smallest integer d such that every kgraph H of order n with minimum l-degree δ l (H) ≥ d contains an F -factor. A classical theorem of Hajnal and Szemerédi [9] implies that t 2 1 (n, K t ) = (1 − 1/t)n for integers t. For k ≥ 3, t k k−1 (n, K k k ) (the δ k−1 (H) threshold for perfect matchings) has been determined by Kühn and Osthus [17] (asymptotically) and Rödl, Ruciński and Szemerédi [24] (exactly) for large n.In this paper, we generalise the absorption technique of Rödl, Ruciński and Szemerédi [24] to F -factors. We determine the asymptotic values of t k 1 (n, K k k (m)) for k = 3, 4 and m ≥ 1. In addition, we show that for t > k = 3 and γ > 0, t 3 2 (n, K 3 t ) ≤ 1 − 2 * Lemma 1.1 (Absorption lemma for F -factors). Let t and i be positive integers and let η > 0. Let F be a hypergraph of order t. Then, there is an integer n 0 = n 0 (t, i, η) satisfying the following: Suppose that H is an (F, i, η)-closed hypergraph of order n ≥ n 0 . Then there exists a vertex subset U ⊆ V (H) of size |U | ≤ (η/2) t n/(4it(t − 1)) with |U | ∈ tZ such that there exists an F -factor in H[U ∪ W ] for every vertex set W ⊆ V \ U of size |W | ≤ (η/2) 2t n/(32i 2 t(t − 1) 2 ) with |W | ∈ tZ.Note that in the above lemma H and F are not necessarily kgraphs, but we only consider k-graphs in this paper. When we say that H has an almost F -factor T , we mean that T is a set of vertexdisjoint copies of F in H such that |V (H) \ V (T )| < ε|V (H)| for some small ε > 0. Equipped with the absorption lemma, we can break down the task of finding an F -factor in large hypergraphs H into the following algorithm.Algorithm for finding F -factors. Remove a small set T 1 of vertex-disjoint copies of F from H such that the resultant graph Hfor some integer i and constant η > 0. 2. Find a vertex set U ⊆ V (H 1 ) satisfying the conditions of the absorption lemma. Set H 2 = H 1 [V (H 1 ) \ U ]. 3. Show that H 2 contains an almost F -factor, i.e. a set T 2 of vertexdisjoint copies of F such that |V (H 2 ) \ V (T 2 )| < ε|V (H 2 )| for small ε > 0.

We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system as a precursor to a quantum theory of gravity. The model has no a priori geometric properties; instead, locality is inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom. We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in this simple model.

In order to gain a better understanding of the Ising model in higher dimensions we have made a comparative study of how the boundary, open versus cyclic, of a d-dimensional simple lattice, for d = 1, . . . , 5, affects the behaviour of the specific heat C and its microcanonical relative, the entropy derivative −∂S/∂U .In dimensions 4 and 5 the boundary has a strong effect on the critical region of the model and for cyclic boundaries in dimension 5 we find that the model displays a quasi first order phase transition with a bimodal energy distribution. The latent heat decreases with increasing systems size but for all system sizes used in earlier papers the effect is clearly visible once a wide enough range of values for K is considered.Relations to recent rigorous results for high dimensional percolation and previous debates on simulation of Ising models and gauge fields are discussed.

For the set of graphs with a given degree sequence, consisting of any number of 2 ′ s and 1 ′ s, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of m-matchings.We find the expected value of the number of m-matchings of r-regular bipartite graphs on 2n vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for m-matchings in r-regular bipartite graphs on 2n vertices, and their asymptotic versions for infinite r-regular bipartite graphs. We prove these conjectures for 2-regular bipartite graphs and for m-matchings with m ≤ 4.2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20 Keywords and phrases: Partial matching and asymptotic growth of average matchings for r-regular bipartite graphs, asymptotic matching conjectures.1 The object of this paper is two folds. First we consider the family Ω(n, k), the set of simple graphs on n vertices with 2k vertices of degree 1 and n − 2k vertices of degree 2. Let Ω bi (n, k) ⊂ Ω(n, k) be the subset of bipartite graphs. For each m ∈ [2, n] ∩ N we characterize the optimal graphs which maximize and minimize φ(m, G), m ≥ 2 for G ∈ Ω(n, k) and G ∈ Ω bi (n, k). It turns out the optimal graphs do not depend on m but on n and k. Furthermore, the graphs with the maximal number of m-matchings, are bipartite.Second, we consider G(2n, r), the set of simple bipartite r-regular graphs on 2n vertices, where n ≥ r. Denote by C l a cycle of length l and by K r,r the complete bipartite graph with r-vertices in each group. For a nonnegative integer q and a graph G denote by qG the disjoint union of q copies of G. Let λ(m, n, r) := min G∈G(2n,r) φ(m, G), Λ(m, n, r) := max G∈G(2n,r) φ(m, G), m = 1, . . . , n. (1.1) Our results on 2-regular graphs yield. λ(m, n, 2) = φ(m, C 2n ), (1.2) Λ(m, 2q, 2) = φ(m, qK 2,2 ), Λ(m, 2q + 3, 2) = φ(m, qK 2,2 ∪ C 6 ), (1.3)for m = 1, . . . , n.The equality Λ(m, 2q, 2) = φ(m, qK 2,2 ) inspired us to conjecture the Upper Matching Conjecture, abbreviated here as UMC: Λ(m, qr, r)) = φ(m, qK r,r ) for m = 1, . . . , qr.( 1.4) For the value m = qr the UMC follows from Bregman's inequality [1]. For the value r = 3 the UMC holds up to q ≤ 8. The results of [4] support the validity of the above conjecture for r = 3, 4 and large values of n. As in the case r = 2 we conjecture that that for any nonbipartite r-regular graph on 2n vertices φ(m, G) ≤ Λ(m, n, r) for m = 1, . . . , n.It is useful to consider G mult (2n, r) ⊃ G(2n, r), the set of r-regular bipartite graphs on 2n vertices, where multiple edges are allowed. Observe that G mult (2, r) = {H r }, where H r is the r-regular multi-bipartite graph on 2 vertices. Let µ(m, n, r) := min G∈G mult (2n,r) φ(m, G), M (m, n, r) := max G∈G mult (2n,r) φ(m, G), (1.5) m = 1, . . . , n, 2 ≤ r ∈ N.It is straightforward to show that M (m, n, r) = φ(m, nH r ) = n m r m , m = 1, . . . , n.(1.6)Hence for most of the values of m Λ(m, n, r) < M (m, n, r). On the other hand, as in the case of Ω(n, k), it i...