arXiv subjects

Grahame Erskine

Publications and source records attributed to Grahame Erskine.

At least 19 recordsLinked to original sources

arXiv2026

Automorphisms of pentagonal geometries with block size 3

A pentagonal geometry PENT(k,r) is a k-uniform, r-regular partial linear space with the property that the set of points not collinear with a given point x is itself a line in the geometry. The existence spectrum of these geometries is known; we begin the study of automorphisms of these structures in the case k=3. We determine the spectrum of pentagonal geometries PENT(3,r) with either a cyclic or reverse automorphism to be r = 1 or 10 (mod 12). In addition, we show that there are 584 pairwise non-isomorphic PENT(3,7)s and classify them by the orders of their automorphism groups.

math.CO↗

arXiv2026

Vertex-transitive closures of graphs

A vertex-transitive closure of $Γ$ is a vertex-transitive supergraph of $Γ$ on the same vertex set. The vertex-transitive number of a graph $Γ$, denoted by $d_{vt}(Γ)$, is the smallest integer for which there exists a $d_{vt}(Γ)$-regular vertex-transitive closure of $Γ$. In this paper we use various algebraic and combinatorial methods to study vertex-transitive closures of graphs and the associated vertex-transitive number.

math.CO↗

arXiv2026

The general position number of digraphs

The general position number for graphs ask for largest vertex subsets $S$ such that no three vertices are contained on a common shortest path. We examine this problem in the setting of directed graphs. We provide bounds for the general position number of digraphs, show that the problem is NP-complete for oriented graphs, investigate the problem for some important families of digraphs such as circulant digraphs, Kautz digraphs and permutation digraphs, and study the general position numbers obtained from all orientations of an undirected graph.

math.CO↗

arXiv2024

Cycle switching in Steiner triple systems of order 19

Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order $v$ (an $STS(v)$), yielding another $STS(v)$. This relationship may be represented by an undirected graph. An $STS(v)$ admits cycles of lengths $4,6,\ldots,v-7$ and $v-3$. In the particular case of $v=19$, it is known that the full switching graph, allowing switching of cycles of any length, is connected. We show that if we restrict switching to only one of the possible cycle lengths, in all cases the switching graph is disconnected (even if we ignore those $STS(19)$s which have no cycle of the given length). Moreover, in a number of cases we find intriguing connected components in the switching graphs which exhibit unexpected symmetries. Our method utilises an algorithm for determining connected components in a very large implicitly defined graph which is more efficient than previous approaches, avoiding the necessity of computing canonical labellings for a large proportion of the systems.

math.CO↗

arXiv2024

Properties of Steiner triple systems of order 21

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.

math.CO↗

arXiv2023

Mutually avoiding Eulerian circuits

Two Eulerian circuits, both starting and ending at the same vertex, are avoiding if at every other point of the circuits they are at least distance 2 apart. An Eulerian graph which admits two such avoiding circuits starting from any vertex is said to be doubly Eulerian. The motivation for this definition is that the extremal Eulerian graphs, i.e. the complete graphs on an odd number of vertices and the cycles, are not doubly Eulerian. We prove results about doubly Eulerian graphs and identify those that are the `densest' and `sparsest' in terms of the number of edges.

math.CO↗

arXiv2022

On the Vertex Position Number of Graphs

In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.

math.CO↗

arXiv2022

Good point sequencings of Steiner triple systems

An l-good sequencing of a Steiner triple system of order v, STS(v), is a permutation of the points of the system such that no l consecutive points in the permutation contains a block. It is known that every STS(v) with v > 3 has a 3-good sequencing. It is proved that every STS(v) with v >= 13 has a 4-good sequencing and every 3-chromatic STS(v) with v >= 15 has a 5-good sequencing. Computational results for Steiner triple systems of small order are also given.

math.CO↗

arXiv2022

Clique-partitioned graphs

A graph $G$ of order $nv$ where $n\geq 2$ and $v\geq 2$ is said to be weakly $(n,v)$-clique-partitioned if its vertex set can be decomposed in a unique way into $n$ vertex-disjoint $v$-cliques. It is strongly $(n,v)$-clique-partitioned if in addition, the only $v$-cliques of $G$ are the $n$ cliques in the decomposition. We determine the structure of such graphs which have the largest possible number of edges.

math.CO↗

arXiv2022

Turan problems for $k$-geodetic digraphs

A digraph $G$ is \emph{$k$-geodetic} if for any pair of (not necessarily distinct) vertices $u,v \in V(G)$ there is at most one walk of length $\leq k$ from $u$ to $v$ in $G$. In this paper we determine the largest possible size of a $k$-geodetic digraph with given order. We then consider the more difficult problem of the largest size of a strongly-connected $k$-geodetic digraph with given order, solving this problem for $k = 2$ and giving a construction which we conjecture to be extremal for larger $k$. We close with some results on generalised Turán problems for the number of directed cycles and paths in $k$-geodetic digraphs.

math.CO↗

arXiv2022

Small graphs and hypergraphs of given degree and girth

The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques to find new record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.

math.CO↗

arXiv2021

Fragments in symmetric configurations with block size 3

We begin the study of collections of three blocks which can occur in a symmetric configuration with block size 3, $v_3$. Formulae are derived for the number of occurrences of these and it is shown that the triangle, i.e. abf, ace, bcd is a basis. It is also shown that symmetric configurations without triangles exist if and only if $v=15$ or $v \geq 17$. Such configurations containing "many" triangles are also discussed and a complete analysis of the triangle content of those with a cyclic automorphism is given.

math.CO↗

arXiv2021

On networks with order close to the Moore bound

The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the \emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic mixed graph with given minimum undirected and directed degrees; this is a generalisation of the classical degree/girth problem. In this paper we present new bounds on the order of mixed graphs with given diameter or geodetic girth and exhibit new examples of directed and mixed geodetic cages. In particular, we show that any $k$-geodetic mixed graph with excess one must have geodetic girth two and be totally regular, thereby proving an earlier conjecture of the authors.

math.CO↗

arXiv2021

Colouring problems for symmetric configurations with block size 3

The study of symmetric configurations $v_3$ with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known $21_3$ and $22_3$ configurations without a blocking set are unique and make a complete enumeration of all non-isomorphic $20_3$ configurations. We discuss the concept of connectivity in relation to symmetric configurations and complete the determination of the spectrum of 2-connected symmetric configurations without a blocking set. A number of open problems are presented.

math.CO↗

arXiv2019

On the upper embedding of symmetric configurations with block size 3

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.

math.CO↗

arXiv2019

Graphs derived from perfect difference sets

We study a family of graphs with diameter two and asymptotically optimal order for their maximum degree, obtained from perfect difference sets. We show that for all known examples of perfect difference sets, the graph we obtain is isomorphic to one of the Brown graphs, a well-known family of graphs in the degree-diameter problem.

math.CO↗

arXiv2018

On total regularity of mixed graphs with order close to the Moore bound

The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected \emph{edges} and directed \emph{arcs}. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter $k$, maximum undirected degree $\leq r$ and maximum directed out-degree $\leq z$. It is also of interest to find smallest possible $k$-geodetic mixed graphs with minimum undirected degree $\geq r$ and minimum directed out-degree $\geq z$. A simple counting argument reveals the existence of a natural bound, the \emph{Moore bound}, on the order of such graphs; a graph that meets this limit is a \emph{mixed Moore graph}. Mixed Moore graphs can exist only for $k = 2$ and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For $k = 2$, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of López and Miret. We also present partial results for larger $k$. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.

math.CO↗

arXiv2018

Self-dual, self-Petrie-dual and Möbius regular maps on linear fractional groups

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality, self-Petrie-duality and Möbius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5,5). The final section includes an enumeration of the $PSL(2,q)$ maps for $q\le81$ and a list of all the $PSL(2,q)$ maps which have any of these special properties for $q\le49$.

math.CO↗