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     

On the number of triangular embeddings of complete graphs and complete tripartite graphs

We prove that for every prime number p and odd m>1, as s→∞, there are at least w face 2‐colorable triangular embeddings of K_{w, w, w}, where w = m·p^s. For both orientable and nonorientable embeddings, this result implies that for infinitely many infinite families of z, there is a constant c>0 for which there are at least z nonisomorphic face 2‐colorable triangular embeddings of K_z. © 2011 Wiley Periodicals, Inc. J Graph Theory     

关于Abel群上Cayley图的Hamilton圈分解

设Ｇ（Ｆ，Ｔ∩Ｔ＾－１）是有限Ａｂｅｌ群Ｆ上的Ｃａｙｌｅｙ图，Ｔ∩Ｔ＾－１只含２阶元，此文证明了当Ｔ是Ｆ的极小生成元集时，若ｄ（Ｇ）＝２ｋ，则Ｇ是ｋ个边不相交的Ｈａｍｉｌｔｏｎ圈的并，若ｄ（Ｇ）＝２ｋ＋１，则Ｇ是ｋ个边不相交的Ｈａｍｉｌｔｏｎ圈与一个１－因子的并。     

Extremal pseudocompact Abelian groups are compact metrizable

Every pseudocompact Abelian group of uncountable weight has both a proper dense pseudocompact subgroup and a strictly finer pseudocompact group topology.

     

Regular Cayley maps for finite abelian groups

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex. Supported in part by the N.Z. Marsden Fund (grant no. UOA0124).     

Self‐embeddings of cyclic and projective Steiner quasigroups

It is shown that for every admissible order v for which a cyclic Steiner triple system exists, there exists a biembedding of a cyclic Steiner quasigroup of order v with a copy of itself. Furthermore, it is shown that for each n≥2 the projective Steiner quasigroup of order 2ⁿ?1 has a biembedding with a copy of itself. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:16‐27, 2010     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.     

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.     

Graphically abelian groups

We define a group G to be graphically abelian if the function g?g⁻¹ induces an automorphism of every Cayley graph of G. We give equivalent characterizations of graphically abelian groups, note features of the adjacency matrices for Cayley graphs of graphically abelian groups, and show that a non-abelian group G is graphically abelian if and only if G=E×Q, where E is an elementary abelian 2-group and Q is a quaternion group.     

On a cardinal group invariant related to decompositions of Abelian groups

For each Abelian groupG, a cardinal invariant χ(G) is introduced and its properties are studied. In the special caseG = ℤ ⁿ , the cardinalχ(ℤ ⁿ ) is equal to the minimal cardinality of an essential subset of ℤ ⁿ , i.e., a of a subsetA ⊂ ℤ ⁿ such that, for any coloring of the group ℤ ⁿ inn colors, there exists an infinite one-color subset that is symmetric with respect to some pointα ofA. The estimaten( ^{n +} l)/2 ≤χ(ℤ ⁿ ) < 2n is proved for alln and the relationχ(ℤ ⁿ ) =n(n + 1)/2 forn ≤ 3. The structure of essential subsets of cardinalityχ(ℤ ⁿ ) in ℤ ⁿ is completely described forn ≤ 3. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 341–350, September, 1998.     

Equitable partitions of Latin‐square graphs

We study equitable partitions of Latin‐square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue ?3.     

Isomorphisms of Cayley graphs of a free Abelian group

A group G is called a CI-group provided that the existence of some automorphism σ ∈ Aut(G) such that σ(A) = B follows from an isomorphism Cay(G, A) ? = Cay (G, B) between Cayley graphs, where A and B are two systems of generators for G. We prove that every finitely generated abelian group is a CI-group.     

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

Generating Nonisomorphic Quadrangular Embeddings of a Complete Graph

We construct (resp. ) index one current graphs with current group such that the current graphs have different underlying graphs and generate nonisomorphic orientable (resp. nonorientable) quadrangular embeddings of the complete graph , (resp. ).     

On mixed Abelian groups with periodic groups of automorphisms

In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established. Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished. Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997. Translated by A. I. Shtern     

Solitary subgroups of Abelian groups

Since solitary subgroups of (infinite) Abelian groups are precisely the strictly invariant subgroups which are co-Hopfian (as groups), and strictly invariant subgroups turn out to be strongly invariant for large classes of Abelian groups we determine the solitary subgroups for these classes of groups.     

Cayley graphs of the group ℤ4 and limits for minimal vertex-primitive graphs of HA-type

We point out a countable set of pairwise nonisomorphic Cayley graphs of the group ℤ⁴ that are limit for finite minimal vertex-primitive graphs admitting a vertex-primitive automorphism group containing a regular Abelian normal subgroup. Supported by RFBR grant No. 06-01-00378. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 203–214, March–April, 2008.