Improved book-embeddings of incomplete hypercubes

In this paper, we show that any incomplete hypercube with, at most, 2ⁿ+2ⁿ⁻¹+2ⁿ⁻² vertices can be embedded in n−1 pages for all n≥4. For the case n≥4, this result improves Fang and Lai’s result that any incomplete hypercube with, at most, 2ⁿ+2ⁿ⁻¹ vertices can be embedded in n−1 pages for all n≥2.Besides this, we show that the result can be further improved when n is large — e.g., any incomplete hypercube with at most 2ⁿ+2ⁿ⁻¹+2ⁿ⁻²+2ⁿ⁻⁷ (respectively, 2ⁿ+2ⁿ⁻¹+2ⁿ⁻²+2ⁿ⁻⁷+2ⁿ⁻²³⁰) vertices can be embedded in n−1 pages for all n≥9 (respectively, n≥232).     

The diameter of a random subgraph of the hypercube

In this paper we present an estimation for the diameter of random subgraph of a hypercube. In the article by A. V. Kostochka (Random Struct Algorithms 4 (1993) 215–229) the authors obtained lower and upper bound for the diameter. According to their work, the inequalities n + m_p ≤ D(G_n) ≤ n + m_p + 8 almost surely hold as n → ∞, where n is dimension of the hypercube and m_p depends only on sampling probabilities. It is not clear from their work, whether the values of the diameter are really distributed on these 9 values, or whether the inequality can be sharpened. In this paper we introduce several new ideas, using which we are able to obtain an exact result: D(G_n) = n + m_p (almost surely). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Biembedding Abelian groups with mates having transversals

A certain recursive construction for biembeddings of Latin squares has played a substantial role in generating large numbers of nonisomorphic triangular embeddings of complete graphs. In this article, we prove that, except for the groups and C ₄ , each Latin square formed from the Cayley table of an Abelian group appears in a biembedding in which the second Latin square has a transversal. Such biembeddings may then be freely used as ingredients in the recursive construction. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:81‐88, 2012     

Latin Squares with Restricted Transversals

The original article to which this erratum refers was correctly published online on 1 December 2011. Due to an error at the publisher, it was then published in Journal of Combinatorial Designs 20: 124–141, 2012 without the required shading in several examples. To correct this, the article is here reprinted in full. The publisher regrets this error. We prove that for all odd there exists a latin square of order 3m that contains an latin subrectangle consisting of entries not in any transversal. We prove that for all even there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order . In particular, we count the number of species of each order that possess an orthogonal mate. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 344–361, 2012     

Existence of self-conjugate self-orthogonal diagonal latin squares

In this article we give some new constructions of self-conjugate self-orthogonal diagonal Latin squares (SCSODLS). As an application of such constructions, we give a conclusive result regarding the existence of SCSODLS and show that there exists an SCSODLS of order n if and only if n ≡ 0, 1 (mod 4), except for n = 5. This result completely disproves a conjecture of Danhof, Phillips, and Wallis about SCSODLS in Danhof, Philips, and Wallis, JCMCC, 8 (1990), 3–8. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:51–62, 1998     

Analytical Expressions for Ising Models on High Dimensional Lattices

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d≥3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For d=5,6,7, our theoretical estimates match the numerical results both qualitatively and quantitatively. For d=3,4, our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d=3), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d=4), the character of the singularity is under current discussion. For the dimensions d=1, 2 the m-vicinity method is not applicable.     

投影均匀分片拉丁超立方体设计

空间填充设计是有效的计算机试验设计,比如均匀设计、最大最小距离拉丁超立方体设计等.虽然这些设计在整个试验空间中有较好的均匀性,但其低维投影均匀性可能并不理想.对于因子是定量的计算机试验,已有文献构造了诸如最大投影设计、均匀投影设计等相适应的设计;而对于同时含有定性因子和定量因子的计算机试验,尚未有投影均匀设计的相关文献.文章提出了综合投影均匀准则,利用门限接受算法构造了投影均匀的分片拉丁超立方体设计.在新构造设计中,整体设计与每一片设计均具有良好的投影均匀性.模拟结果显示,与随机分片拉丁超立方体设计相比,利用新构造设计进行试验而拟合的高斯过程模型具有更小的均方根预测误差.     

EXISTENCE OF THREE-FOLD BIBDs WITH BLOCK-SIZE SEVEN

Hanani discussed the existence of balanced incomplete block designs B[7,3;ν] leaving seven open cases of v , five of which were solved by Greig and Yin and Wu. In this paper we shall deal with the last two cases and then complete the existence question.     

Tabu Search for the planar three-index assignment problem

Tabu Search is a very effective method for approximately solving hard combinatorial problems. In the current work we implement Tabu Search for solving the Planar Three-Index Assignment Problem. The problem deals with finding the minimum cost assignment between elements of three distinct sets demanding that every pair of elements, each representing a different set, appears in the same solution exactly once. The solutions of the problems correspond to Latin squares. These structures form the basis of the move generation mechanism employed by the Tabu Search procedures. Standard Tabu Search ideas such as candidate move list, variable tabu list size, and frequency-based memory are tested. Computational results for a range of problems of varying dimensions are presented.