Journal of Combinatorial Designs最新文献

We generalize the well-known Oberwolfach problem posed by Ringel in 1967. We suppose to have $��\frac{v}{2}�$ couples (here $��v�\; �\ge �\; �4�$ is an even integer) and suppose that they have to be seated for several nights at $��t�$ round tables in such a way that each person seats next to his partner exactly $��r�\; �\ge �\; �0�$ times and next to every other person exactly once. We call this problem the Oberwolfach problem with loving couples. When $��r�\; �=�\; �0�$, the problem coincides with the so-called spouse-avoiding variant, which was introduced by Huang, Kotzig, and Rosa in 1979. While if either $��r�\; �=�\; �2�$ or $��r�$ equals the number of nights, it corresponds to the spouse-loving variant or to the Honeymoon variant, which was recently studied by Bolohan et al. and by Lepine and Sajna, respectively. In this paper, for each possible choice of $��r�$, we construct many classes of solutions to the Oberwolfach problem with loving couples. We also obtain new solutions to the Honeymoon variant.

A multifold 1-perfect code (1-perfect code for list decoding) in any graph is a set $�\; ��\; �C��$ of vertices such that every vertex of the graph is at distance not more than 1 from exactly $�\; ��\; �\mu ��$ elements of $�\; ��\; �C��$. In $�\; ��\; �q��$-ary Hamming graphs, where $�\; ��\; �q��$ is a prime power, we characterize all parameters of multifold 1-perfect codes and all parameters of additive multifold 1-perfect codes. In particular, we show that additive multifold 1-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold 1-perfect codes always exist.

This paper is a contribution to the classification of all pairs $��\left(��T�\; �,�\; �G��\; �\right)�$, where $��T�$ is a triple system and $��G�$ is a block-transitive but not flag-transitive automorphism group of $��T�$. We prove that if the socle of $��G�$ is a sporadic or alternating group, then one of the following holds:

In 1991, Shalaby conjectured that any $�\; ��Z_n�$, where $�\; ��n�\; �\equiv �\; �1�$ or $�\; ��3��\; ��\left(�\; ��\mathrm{mod}��\; �8��\; �\right)��\; �,�\; �n�\; �\ge �\; �11�$, admits a strong Skolem starter. In 2018, the authors fully described and explicitly constructed the infinite “cardioidal” family of strong Skolem starters. No other infinite family of these combinatorial designs was known to date. Statements regarding the products of starters, proven in this paper give a new way of generating strong or skew Skolem starters of composite orders. This approach extends our previous result by generating new infinite families of these starters that are not cardioidal.

Orthogonal array and a large set of orthogonal arrays are important research objects in combinatorial design theory, and they are widely applied to statistics, computer science, coding theory, and cryptography. In this paper, some new series of large sets of orthogonal arrays are given by direct construction, juxtaposition construction, Hadamard construction, finite field construction, and difference matrix construction. Subsequently, many new infinite classes of orthogonal arrays are obtained by using these large sets of orthogonal arrays and Kronecker product.

For any finite group $�\; ��G�$, the Terwilliger algebra $�\; ��T�\; ��\left(�\; �G�\; �\right)��$ of the group association scheme satisfies the following inclusions: $�\; ��T_0�\; ��\left(�\; �G�\; �\right)��\; �\subseteq �\; �T�\; ��\left(�\; �G�\; �\right)��\; �\subseteq �\; �\tilde{T}�\; ��\left(�\; �G�\; �\right)��$, where $�\; ��T_0�\; ��\left(�\; �G�\; �\right)��$ is a specific vector space and $�\; ��\tilde{T}�\; ��\left(�\; �G�\; �\right)��$ is the centralizer algebra of the permutation representation of $�\; ��G�$ induced by the action of conjugation. The group $�\; ��G�$ is said to be triply transitive if $�\; ��T_0�\; ��(�\; �G�$

Let $�\; ��V�$ be a vector space over the finite field $�\; ��F_q�$. A $�\; ��q�$-Steiner system, or an $�\; ��S�\; �{�\left(�\; ��t�\; �,�\; �k�\; �,�\; �V��\; �\right)�}_q�$, is a collection $�\; ��ℬ�$ of $�\; ��k�$-dimensional subspaces of $�\; ��V�$ such that every $�\; ��t�$-dimensional subspace of $�\; ��V�$ is contained in a unique element of $�\; ��ℬ�$

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.