Grahame Erskine

The Open University

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James Tuite

The Open University

Terry S. Griggs

The Open University

JŠ

Jozef Širáň

Slovak University of Technology in Bratislava

Katarína Hriňáková

Slovak University of Technology in Bratislava

M.A. Fiol

Universitat Politècnica de Catalunya

Geoffrey Exoo

Indiana State University

Olivia Jeans

The Open University

Nika Salia

King Fahd University of Petroleum and Minerals

David Bevan

University of Strathclyde

C. Dalfó

Universitat de Lleida

Cooperative Institutions

The Open University

University of Kerala

Universitat de Lleida

Hong Kong Metropolitan University

Universidade Aberta

Universitat Politècnica de Catalunya

University of Newcastle Australia

University of L'Aquila

University of Maribor

Institute of Mathematics, Physics, and Mechanics

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On the upper embedding of symmetric configurations with block size 3

Discrete Mathematics (2019)

Grahame Erskine Terry S. Griggs Jozef Širáň

Block size

Table (database)

10.1016/j.disc.2019.111774

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Citations (4)

Developing STACK practice questions for the Mathematics Masters Programme at the OU

MSOR Connections (2018)

Ben Mestel Grahame Erskine

The design and implementation of Masters level e-assessment in STACK is described for the introductory module M820 Calculus of Variations and Advanced Calculus in the Open University’s Masters Programme in Mathematics. Some basic design principles are described and illustrated for an online practice quiz on the use of the Jacobi equation to classify stationary paths.

Mathematical practice

10.21100/msor.v17i1.896

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Citations (2)

Graphs derived from perfect difference sets

Australas. J Comb. (2021)

Grahame Erskine Peter Fratrič Jozef Širáň

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.

Degree (music)

Indifference graph

Trivially perfect graph

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Citations (1)

Large circulant graphs of fixed diameter and arbitrary degree

Ars Mathematica Contemporanea (2017)

David Bevan Grahame Erskine Robert R. Lewis

We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for “stitching” together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum size of a subset A of a cyclic group of order n such that the k-fold sumset kA is equal to the whole group. We also present a revised table of largest known circulant graphs of small degree and diameter.

Degree (music)

Image stitching

10.26493/1855-3974.969.659

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Citations (6)

On affine forestry over integral domains and families of deep Jordan–Gauss graphs

European Journal of Mathematics (2025)

Tymoteusz Chojecki Grahame Erskine James Tuite Vasyl Ustimenko

10.1007/s40879-024-00798-2

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Cycle switching in Steiner triple systems of order 19

arXiv (Cornell University) (2024)

Grahame Erskine Terry S. Griggs

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.

10.48550/arxiv.2405.07750

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Tur\'{a}n problems for $k$-geodetic digraphs

arXiv (Cornell University) (2021)

James Tuite Grahame Erskine Nika Salia

Turan-type problems have been widely investigated in the context of undirected simple graphs. Turan problems for paths and cycles in directed graphs have been treated by Bermond, Heydemann et al. Recently $k$-geodetic digraphs have received great attention in the study of extremal problems, particularly in a directed analogue of the degree/girth problem. Ustimenko et al studied the problem of the largest possible size of a diregular $k$-geodetic digraph with order $n$ and gave asymptotic estimates. In this paper we consider the maximal size of a $k$-geodetic digraph with given order with no assumption of diregularity. We solve this problem for all $k$ and classify the extremal digraphs. We then provide a partial solution for the more difficult problem of the largest number of arcs in a strongly-connected $k$-geodetic digraph with given order and provide lower bounds for these numbers that we conjecture to be extremal for sufficiently large $n$. We close with some results on generalised Turan problems for $k$-geodetic digraphs.

Digraph

Source

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Colouring problems for symmetric configurations with block size 3

Journal of Combinatorial Designs (2021)

Grahame Erskine Terry S. Griggs Jozef Širáň

Abstract The study of symmetric configurations 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 and configurations without a blocking set are unique and make a complete enumeration of all nonisomorphic 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.

Blocking (statistics)

Enumeration

Block size

Monochromatic color

10.1002/jcd.21773

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Citations (1)

New Bounds on k-Geodetic Digraphs and Mixed Graphs

Trends in mathematics (2021)

James Tuite Grahame Erskine

Digraph

Mixed graph

Degree (music)

Undirected graph

10.1007/978-3-030-83823-2_124

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Small Graphs and Hypergraphs of Given Degree and Girth

The Electronic Journal of Combinatorics (2023)

Grahame Erskine James Tuite

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.

Hypergraph

Girth (graph theory)

10.37236/11765

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Citations (2)