Related papers: On $m$-partite oriented semiregular representation…

Let $G$ be a finite group and $m \geq 2$ a positive integer. We say that $G$ admits an \emph{oriented $m$-semiregular representation} (abbreviated as OmSR) if there exists a $m$-Cayley digraph $\Gamma$ over $G$ such that $\Gamma$ is…

Extending the well-studied concept of graphical regular representations to bipartite graphs, a Haar graphical representation (HGR) of a group $G$ is a bipartite graph whose automorphism group is isomorphic to $G$ and acts semiregularly with…

This is the third, and last, of a series of papers dealing with oriented regular representations. Here we complete the classification of finite groups that admit an oriented regular representation (or ORR for short), and give a complete…

In this paper we give a partial answer to a 1980 question of Lazslo Babai: "Which [finite] groups admit an oriented graph as a DRR?" That is, which finite groups admit an oriented regular representation (ORR)? We show that every finite…

Given a group $G$, an {\em $m$-Cayley digraph $\G$ over $G$} is a digraph that has a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. We say that $G$ admits an {\em oriented $m$-semiregular…

In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called…

A finite group $G$ admits an {\em oriented regular representation} if there exists a Cayley digraph of $G$ such that it has no digons and its automorphism group is isomorphic to $G$. Let $m$ be a positive integer. In this paper, we extend…

Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $\Gamma$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $\Sigma$…

Let $R$ be a group and let $S$ be a subset of $R$. The Haar graph $\mathrm{Haar}(R,S)$ of $R$ with connection set $S$ is the graph having vertex set $R\times\{-1,1\}$, where two distinct vertices $(x,-1)$ and $(y,1)$ are declared to be…

This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible…

Let $K_{m[n]}$ be the complete multipartite graph with $m$ parts, while each part contains $n$ vertices. The orientably-regular embeddings of complete graphs $K_{m[1]}$ have been determined by Biggs (1971) \cite{Big1}, James and Jones…

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR. First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete…

In this paper we extend the notion of digraphical regular representations in the context of Haar digraphs. Given a group $G$, a {\em Haar digraph} $\Gamma$ over $G$ is a bipartite digraph having a bipartition $\{X,Y\}$ such that $G$ is a…

We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…

By a result of Babai, with finitely many exceptions, every group $G$ admits a semi-regular poset representation with three orbits, that is, a poset $P$ with automorphism group $\textrm{Aut}(P) \simeq G$ such that the action of…

An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge…

For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $\Gamma$ such that the automorphism group of $\Gamma$ is isomorphic to $G$ and acts…

The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum--Weber, 2015) and skew categories of…

Let $G$ be a finite group and $m$ be an integer. We employ the notation $g_i$ to represent elements $(g,i)$ in the Cartesian product $G \times \mathbb{Z}_m$, where $\mathbb{Z}_m$ denotes integers modulo $m$. For given sets $T_{i,j}…

In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a…