All λ‐Designs with Small λ are Type‐1

A λ‐design is a family of subsets of such that for all and not all are of the same size. Ryser's and Woodall's λ‐design conjecture states that each λ‐design can be obtained from a symmetric block design by a certain complementation procedure. Our main result is that the conjecture is true when λ < 63. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 408–431, 2012     

Symmetries that latin squares inherit from 1‐factorizations

A 1‐factorization of a graph is a decomposition of the graph into edge disjoint perfect matchings. There is a well‐known method, which we call the ??‐construction, for building a 1‐factorization of K_n,n from a 1‐factorization of K_{n + 1}. The 1‐factorization of K_n,n can be written as a latin square of order n. The ??‐construction has been used, among other things, to make perfect 1‐factorizations, subsquare‐free latin squares, and atomic latin squares. This paper studies the relationship between the factorizations involved in the ??‐construction. In particular, we show how symmetries (automorphisms) of the starting factorization are inherited as symmetries by the end product, either as automorphisms of the factorization or as autotopies of the latin square. Suppose that the ??‐construction produces a latin square L from a 1‐factorization F of K_{n + 1}. We show that the main class of L determines the isomorphism class of F, although the converse is false. We also prove a number of restrictions on the symmetries (autotopies and paratopies) which L may possess, many of which are simple consequences of the fact that L must be symmetric (in the usual matrix sense) and idempotent. In some circumstances, these restrictions are tight enough to ensure that L has trivial autotopy group. Finally, we give a cubic time algorithm for deciding whether a main class of latin squares contains any square derived from the ??‐construction. The algorithm also detects symmetric squares and totally symmetric squares (latin squares that equal their six conjugates). © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 157–172, 2005.     

A note on 3‐factorizations of K10*

In this note, we enumerate all nonisomorphic 3‐factorizations of K₁₀. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 379–383, 2001     

The Existence of Partitioned Generalized Balanced Tournament Designs with Block Size 3

A generalized balanced tournament design, GBTD(n, k), defined on a kn-set V, is an arrangement of the blocks of a (kn, k, k – 1)-BIBD defined on V into an n × (kn – 1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. Suppose we can partition the columns of a GBTD(n, k) into k + 1 sets B1, B2,..., Bk + 1 where |Bi| = n for i = 1, 2,..., k – 2, |Bi| = n–1 for i = k – 1, k and |Bk+1| = 1 such that (1) every element of V occurs precisely once in each row and column of Bi for i = 1, 2,..., k – 2, and (2) every element of V occurs precisely once in each row and column of Bi Bk+1 for i = k – 1 and i = k. Then the GBTD(n, k) is called partitioned and we denote the design by PGBTD(n, k). The spectrum of GBTD(n, 3) has been completely determined. In this paper, we determine the spectrum of PGBTD(n,3) with, at present, a fairly small number of exceptions for n. This result is then used to establish the existence of a class of Kirkman squares in diagonal form.     

Embedding Factorizations for 3‐Uniform Hypergraphs

In this article, two results are obtained on a hypergraph embedding problem. The proof technique is itself of interest, being the first time amalgamations have been used to address the embedding of hypergraphs. The first result finds necessary and sufficient conditions for the embedding a hyperedge‐colored copy of the complete 3‐uniform hypergraph of order m, , into an r‐factorization of , providing that . The second result finds necessary and sufficient conditions for an embedding when not only are the colors of the hyperedges of given, but also the colors of all the “pieces” of hyperedges on these m vertices are prescribed (the “pieces” of hyperedges are eventually extended to hyperedges of size 3 in by adding new vertices to the hyperedges of size 1 and 2 during the embedding process). Both these results make progress toward settling an old question of Cameron on completing partial 1‐factorizations of hypergraphs. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 216–224, 2013     

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

Given two 2‐regular graphs F₁ and F₂, both of order n, the Hamilton‐Waterloo Problem for F₁ and F₂ asks for a factorization of the complete graph into α₁ copies of F₁, α₂ copies of F₂, and a 1‐factor if n is even, for all nonnegative integers α₁ and α₂ satisfying . We settle the Hamilton‐Waterloo Problem for all bipartite 2‐regular graphs F₁ and F₂ where F₁ can be obtained from F₂ by replacing each cycle with a bipartite 2‐regular graph of the same order.     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Block‐Transitive and Point‐Primitive 2‐ Designs with Sporadic Socle

The purpose of this paper is to classify all pairs , where is a nontrivial 2‐ design, and acts transitively on the set of blocks of and primitively on the set of points of with sporadic socle. We prove that there exists only one such pair : is the unique 2‐(176,8,2) design and , the Higman–Sims simple group.     

Enclosings of decompositions of complete multigraphs in 2‐factorizations

Let k, m, n, λ, and μ be positive integers. A decomposition of into edge‐disjoint subgraphs is said to be enclosed by a decomposition of into edge‐disjoint subgraphs if and, after a suitable labeling of the vertices in both graphs, is a subgraph of and is a subgraph of for all . In this paper, we continue the study of enclosings of given decompositions by decompositions that consist of spanning subgraphs. A decomposition of a graph is a 2‐factorization if each subgraph is 2‐regular and spanning, and is Hamiltonian if each subgraph is a Hamiltonian cycle. We give necessary and sufficient conditions for the existence of a 2‐factorization of that encloses a given decomposition of whenever and . We also give necessary and sufficient conditions for the existence of a Hamiltonian decomposition of that encloses a given decomposition of whenever and either or and , or , , and .     

Some new results on 1‐rotational 2‐factorizations of the complete graph

It is known that a necessary condition for the existence of a 1‐rotational 2‐factorization of the complete graph K_2n+1 under the action of a group G of order 2n is that the involutions of G are pairwise conjugate. Is this condition also sufficient? The complete answer is still unknown. Adapting the composition technique shown in Buratti and Rinaldi, J Combin Des, 16 (2008), 87–100, we give a positive answer for new classes of groups; for example, the groups G whose involutions lie in the same conjugacy class and having a normal subgroup whose order is the greatest odd divisor of |G|. In particular, every group of order 4t+2 gives a positive answer. Finally, we show that such a composition technique provides 2‐factorizations with a rich group of automorphisms. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 237–247, 2010     

Generalized Bhaskar Rao Designs with Block Size 3 over Finite Abelian Groups

We show that if G is a finite Abelian group and the block size is 3, then the necessary conditions for the existence of a (v,3,λ;G) GBRD are sufficient. These necessary conditions include the usual necessary conditions for the existence of the associated (v,3,λ) BIBD plus λ≡ 0 (mod|G|), plus some extra conditions when |G| is even, namely that the number of blocks be divisible by 4 and, if v = 3 and the Sylow 2-subgroup of G is cyclic, then also λ≡ 0 (mod2|G|).     

Flag‐Transitive 2‐ Symmetric Designs with Sporadic Socle

Let be a nontrivial 2‐ symmetric design admitting a flag‐transitive, point‐primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2‐(144, 66, 30) design with or , a 2‐(176, 50, 14) design with , a 2‐(176, 126, 90) design with or , or a 2‐(14,080, 12,636, 11,340) design with .     

The Hamilton–Waterloo Problem for ‐Factors and ‐Factors

The Hamilton–Waterloo problem asks for a 2‐factorization of (for v odd) or minus a 1‐factor (for v even) into ‐factors and ‐factors. We completely solve the Hamilton–Waterloo problem in the case of C₃‐factors and ‐factors for .     

2‐Distance Coloring of Sparse Graphs

For graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2‐distance ‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance ‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than and with large enough maximum degree Δ (depending only on m) can be list 2‐distance ‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance ‐colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2‐distance ‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less.     

给定密度随机数的生成的通用方法与球面和球体上均匀随机向量的生成法

本文首先介绍基于垂直概率密度表示的,给定密度函数的随机数生成的通用方法;然后介绍球面及球体上均匀随机向量的生成算法。     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

The asphericity of random 2‐dimensional complexes

We study random 2‐dimensional complexes in the Linial–Meshulam model and prove that for the probability parameter satisfying a random 2‐complex Y contains several pairwise disjoint tetrahedra such that the 2‐complex Z obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex Z satisfies the Whitehead conjecture, i.e. any subcomplex is aspherical. This implies that Y is homotopy equivalent to a wedge where Z is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions where c > 3 and , the complex Z is genuinely 2‐dimensional and in particular, it has sizable 2‐dimensional homology; it follows that in the indicated range of the probability parameter p the cohomological dimension of the fundamental group of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015     

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2‐connected (simple) graphs on n vertices and m edges, provided that m ‐ n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2‐edge‐connectedness is treated similarly. We also obtain formulae for the number of 2‐connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013