Simple constructions for balanced incomplete block designs with block size three

AbstractLet S be a finite set with v elements. It is known that there exists a sequence of three-element subsets of S such that each two-element subset of S is contained in exactly λ terms of the sequence if and only if λ(v − 1)2 and λv(v − 1)6 are integers. The known proof is somewhat complicated when v ≡ 2 (mod 6), and this paper provides a simpler proof for this case. Proofs are also given for all other values of v by reviewing known constructions or providing new ones

Mixing time of critical Ising model on trees is polynomial in the height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature $\beta_c$, the inverse-gap is bounded for $\beta < \beta_c$, polynomial in the surface area for $\beta = \beta_c$ and exponential in it for $\beta > \beta_c$. This has been proved for $\Z^2$ except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for $\beta \beta_c$ were established, where $\beta_c$ is the critical spin-glass parameter, and the tree-height $h$ plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the $b$-ary tree, by showing that it is indeed polynomial in $h$ at criticality. The degree of our polynomial bound does not depend on $b$, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for $\beta > \beta_c$, the inverse-gap and mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.Comment: 53 pages; 3 figure

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

Infinite graphs—A survey

AbstractThis expository article describes work which has been done on various problems involving infinite graphs, mentioning also a few unsolved problems or suggestions for future investigation

Hamiltonian double latin squares

AbstractA double latin square of order 2n on symbols σ1,…,σn is a 2n×2n matrix A=(aij) in which each aij is one of the symbols σ1,…,σn and each σk occurs twice in each row and twice in each column. For k=1,…,n let B(A,σk) be the bipartite graph with vertices ρ1,…,ρ2n,c1,…,c2n and 4n edges [ρi,cj] corresponding to ordered pairs (i,j) such that aij=σk. We say that A is Hamiltonian if B(A,σk) is a cycle of length 4n for k=1,…,n. Two double latin squares (aij),(aij′) of order 2n on symbols σ1,…,σn are said to be orthogonal if for each ordered pair (σh,σk) of symbols there are four ordered pairs (i,j) such that aij=σh, aij′=σk.We explore ways of constructing Hamiltonian double latin squares (HLS), symmetric HLS, sets of mutually orthogonal HLS and pairs of orthogonal symmetric HLS. We identify those arrays which can be obtained from HLS by amalgamating rows and amalgamating columns in a certain sense, and we prove a similar result concerning symmetric arrays obtainable in this way from symmetric HLS. These results can be proved either by using matroids or by a more elementary method, and we illustrate both approaches. From these results we deduce a characterisation of those matrices which are submatrices of HLS on n symbols, a similar result concerning symmetric submatrices of symmetric HLS and some related results. Much of our discussion uses graph-theoretic language, since HLS on n symbols are equivalent to decompositions of K2n,2n into Hamiltonian cycles and symmetric HLS on n symbols are equivalent to decompositions of K2n into Hamiltonian paths (and these are equivalent to decompositions of K2n+1 into Hamiltonian cycles)

Continuous Fraïssé Conjecture

We investigate the relation of countable closed linear orderings with respect to continuous monotone embeddability and show that there are exactly ℵ_1 many equivalence classes with respect to this embeddability relation. This is an extension of Laver's result, who considered (plain) embeddability, which yields coarser equivalence classes. Using this result we show that there are only ℵ_0 many different Gödel logics

Combined connectivity augmentation and orientation problems

Two important branches of graph connectivity problems are connectivity augmentation, which consists of augmenting a graph by adding new edges so as to meet a specified target connectivity, and connectivity orientation, where the goal is to find an orientation of an undirected or mixed graph that satisfies some specified edge-connection property. In the present work an attempt is made to link the above two branches, by considering degree-specified and minimum cardinality augmentation of graphs so that the resulting graph admits an orientation satisfying a prescribed edge-connection requirement, such as (k,l)-edge-connectivity. The results are obtained by combining the supermodular polyhedral methods used in connectivity orientation with the splitting off operation, which is a standard tool in solving augmentation problems