arXiv subjects

James Tuite

Publications and source records attributed to James Tuite.

30 records · Page 2Linked to original sources

arXiv2022

On monophonic position sets in graphs

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 prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.

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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.

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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.

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arXiv2022

On some extremal position problems for graphs

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 discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers. We then determine the asymptotic order of the largest size of a graph with given general or monophonic position number, classifying the extremal graphs with monophonic position number two. Finally we establish the possible diameters of graphs with given order and monophonic position number.

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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.

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arXiv2021

The structure of digraphs with excess one

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 trivial cases $d = 1$ and $k = 1$, so it is of interest to look for $k$-geodetic digraphs with out-degree $d$ and smallest possible order $M(d,k)+ε$, where $ε$ is the \emph{excess} of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for $k = 3,4$ and $d \geq 2$ and for $k = 2$ and $d \geq 8$. We conjecture that there are no digraphs with excess one for $d,k \geq 2$ and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the non-existence of certain digraphs with degree three and excess one, as well closing the open cases $k = 2$ and $d = 3,4,5,6,7$ left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, i.e. the outlier function of any such digraph must contain a cycle of length $\geq 3$.

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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.

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arXiv2021

On diregular digraphs with degree two and excess three

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 that between any pair of vertices $u,v$ there is at most one directed path of length $\leq k$ from $u$ to $v$; such a digraph has order $M(d,k)+ε$ for some small excess $ε$. Sillasen et al. have shown that there are no digraphs with out-degree two and excess one. The present author has classified all digraphs with out-degree two and excess two. In this paper it is proven that there are no diregular digraphs with out-degree two and excess three for $k \geq 3$, thereby providing the first classification of digraphs with order three away from the Moore bound for a fixed out-degree.

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arXiv2019

Digraphs with degree two and excess two are diregular

A $k$-geodetic digraph with minimum out-degree $d$ has excess $ε$ if it has order $M(d,k) + ε$, where $M(d,k)$ represents the Moore bound for out-degree $d$ and diameter $k$. For given $ε$, 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-regularity is in general non-trivial. It has recently been shown that any digraph with excess $ε= 1$ must be diregular. In this paper we prove that digraphs with minimum out-degree $d = 2$ and excess $ε= 2$ are diregular for $k \geq 2$.

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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.

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arXiv2018

Large Cayley graphs of small diameter

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 restrict attention to Cayley graphs, and study the asymptotics by fixing a small diameter and constructing families of graphs of large order for all values of the maximum degree. Much of the literature in this direction is focused on the diameter two case. In this paper we consider larger diameters, and use a variety of techniques to derive new best asymptotic constructions for diameters 3, 4 and 5 as well as an improvement to the general bound for all odd diameters. Our diameter 3 construction is, as far as we know, the first to employ matrix groups over finite fields in the degree-diameter problem.

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arXiv2017

On diregular digraphs with degree two and excess two

An important topic in the design of efficient networks is the construction of $(d,k,+ε)$-digraphs, i.e. $k$-geodetic digraphs with minimum out-degree $\geq d$ and order $M(d,k)+ ε$, where $M(d,k)$ represents the Moore bound for degree $d$ and diameter $k$ and $ε> 0$ is the (small) excess of the digraph. Previous work has shown that there are no $(2,k,+1)$-digraphs for $k \geq 2$. In a separate paper, the present author has shown that any $(2,k,+2)$-digraph must be diregular for $k \geq 2$. In the present work, this analysis is completed by proving the nonexistence of diregular $(2,k,+2)$-digraphs for $k \geq 3$ and classifying diregular $(2,2,+2)$-digraphs up to isomorphism.

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