James Tuite

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• PhD Pure Mathematics
• Visiting Fellow at The Open University

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we r...

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 edges and directed arcs. The degree/diameter problem fo...

Tur\'{a}n-type problems have been widely investigated in the context of undirected simple graphs. Tur\'{a}n 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/...

The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but with 'induced path' in place of 'geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we di...

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 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...

Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for the largest sets $S$ of vertices in a graph such that no three elements of $S$ lie on a common shortest path. The number of vertices in such a largest set is the general position number of the graph. This paper provides a survey of this rapidly growing...

Let K be a commutative ring. We refer to a connected bipartite graph \(G=G_n(K)\) with partition sets \(P=K^n\) (points) and \(L=K^n\) (lines) as an affine graph over K of dimension \(\dim \hspace{0.55542pt}(G)=n\) if the neighbourhood of each vertex is isomorphic to K. We refer to G as an algebraic affine graph over K if the incidence between a po...

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs, in terms of the number of vertices, are presented for small diameters. Moreover, two infinite famil...

The general position problem in graph theory asks for the number of vertices in a largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. The analogous monophonic position problem is obtained from the general position problem by replacing ``shortest path" by ``induced path." This paper st...

Let $K$ be a commutative ring. We refer to a connected bipartite graph $G=G_n(K)$ with partition sets $P=K^n$ (points) and $L=K^n$ (lines) as an \emph{affine graph} over $K$ of dimension $\dim(G)=n$ if the neighbourhood of each vertex is isomorphic to $K$. We refer to $G$ as an \emph{algebraic affine graph} over $K$ if the incidence between a point...

In this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour $V(G)$ such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic num...

A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as ∑i≥0aixi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage...

A digraph is ‐ geodetic if for any (not necessarily distinct) vertices there is at most one directed walk from to with length not exceeding . The order of a ‐geodetic digraph with minimum out‐degree is bounded below by the directed Moore bound . The Moore bound can be met only in the trivial cases and , so it is of interest to look for ‐geodetic di...

A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as \sum_{i \geq 0} a_i x^i, where a_i is the number of distinct general position sets of G with cardinality i. The polynomial is considered for several well-...

A subset S of vertices of a graph G is in general position if no shortest path in G contains three vertices of S. The general position problem asks for the largest number of vertices in a general position set of G, whilst the lower general position problem asks for a smallest maximal general position set. In this paper, we discuss the lower general...

The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we di...

A subset S of vertices of a graph G is a general position set if no shortest path in G contains three or more vertices of S. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the lower general position number gp − (G) of G, which is the number of vertices in a smallest maximal general position set of G. We show tha...

An (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in-and out-degree z, and diameter k. In this paper, we study the case r = z = 1, proposing some new constructions of (1, 1, k)-mixed graphs with a large number of vertices N. Our study is based on computer techniques for small values of k and the use of graphs on alphabe...

An $(r,z,k)$-mixed graph $G$ has every vertex with undirected degree $r$, directed in- and out-degree $z$, and diameter $k$. In this paper, we study the case $r=z=1$, proposing some new constructions of $(1,1,k)$-mixed graphs with a large number of vertices $N$. Our study is based on computer techniques for small values of $k$ and the use of graphs...

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. t...

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 v...

The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected n-vertex k-uniform hypergraph and characterize for every n all hypergraphs attaining the maximum Wiener index.

A digraph G is k-geodetic if for any pair of (not necessarily distinct) vertices u,v∈V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u,v \in V(G)$$\end{document} th...

Let G be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited while maintaining the general position property...

Let G be a graph. Assume that to each vertex of a set of vertices S ⊆ V (G) a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited whilst maintaining the general position property at all...

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 ⊆ V (G) is an x-position set if for any y ∈ S the shortest x, y-paths in G contain no point of S \ {y}. We investigate the largest and smallest orders of maximum x-positio...

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 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 proble...

The general position problem for graphs was inspired by the no-three-in-line problem and the general position subset selection problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path between two vertices of S contains a third element of S; the general position number of G is the size of a large...

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 v...

The general position problem for graphs stems from a puzzle of Dudeney and the general position problem from discrete geometry. The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but wit...

A digraph $G$ is \emph{$k$-geodetic} if for any (not necessarily distinct) vertices $u,v$ there is at most one directed walk from $u$ to $v$ with length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the directed Moore bound $M(d,k) = 1+d+d^2+\dots +d^k$. The Moore bound can be met only in the...

We study a generalisation of the degree/girth problem to the setting of directed and mixed graphs. We say that a mixed graph or digraph G is k-geodetic if there is no pair of vertices u, v such that G contains distinct non-backtracking walks of length ≤k from u to v. The order of a k-geodetic mixed graph with minimum undirected degree r and minimum...

The general position problem in graph theory asks for the largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. In this paper we consider a variant of the general position problem called the \emph{monophonic position problem}, obtained by replacing `shortest path' by `induced path'. We...

A set $S$ of vertices in a graph $G$ is a general position set if no three vertices of $S$ lie on a common geodesic path in $G$. The size of the largest general position set of $G$ is called the general position number of $G$, denoted by ${\rm gp}(G).$ A monophonic path $P$ in $G$ is a path in which any two non-consecutive vertices are not connecte...

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/diame...

Moore digraphs, that is digraphs with out-degree $d$, diameter $k$ and order equal to the Moore bound $M(d,k) = 1 + d + d^2 + \dots +d^k$, arise in the study of optimal network topologies. In an attempt to find digraphs with a 'Moore-like' structure, attention has recently been devoted to the study of small digraphs with minimum out-degree $d$ such...

Moore digraphs, that is digraphs with out-degree $d$, diameter $k$ and order equal to the Moore bound $M(d,k) = 1 + d + d^2 + \dots +d^k$, arise in the study of optimal network topologies. In an attempt to find digraphs with a `Moore-like' structure, attention has recently been devoted to the study of small digraphs with minimum out-degree $d$ such...

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we r...

An important topic in the design of efficient networks is the construction of $(d,k,+\epsilon )$-digraphs, i.e. $k$-geodetic digraphs with minimum out-degree $\geq d$ and order $M(d,k)+ \epsilon $, where $M(d,k)$ represents the Moore bound for degree $d$ and diameter $k$ and $\epsilon > 0$ is the (small) excess of the digraph. Previous work has sho...

An important topic in the design of efficient networks is the construction of $(d,k,+\epsilon )$-digraphs, i.e. $k$-geodetic digraphs with minimum out-degree $\geq d$ and order $M(d,k)+ \epsilon $, where $M(d,k)$ represents the Moore bound for degree $d$ and diameter $k$ and $\epsilon > 0$ is the (small) excess of the digraph. Previous work has sho...

A $k$-geodetic digraph with minimum out-degree $d$ has excess $\epsilon $ if it has order $M(d,k) + \epsilon $, where $M(d,k)$ represents the Moore bound for degree $d$ and diameter $k$. For given $\epsilon $, it is simple to show that any such digraph must be out-regular with degree $d$ for sufficiently large $d$ and $k$. However, proving in-regul...

A $k$-geodetic digraph with minimum out-degree $d$ has excess $\epsilon $ if it has order $M(d,k) + \epsilon $, where $M(d,k)$ represents the Moore bound for out-degree $d$ and diameter $k$. For given $\epsilon $, it is simple to show that any such digraph must be out-regular with degree $d$ for sufficiently large $d$ and $k$. However, proving in-r...