arXiv subjects

David A. Pike

Publications and source records attributed to David A. Pike.

At least 19 recordsLinked to original sources

arXiv2026

Burning Steiner triple systems

Graph burning is a round-based process which can be viewed as a discrete one-player game that models the spread of influence throughout a network. Extending this process to hypergraphs can be done in numerous ways; two such processes that have been studied include the burning process and the lazy burning process for hypergraphs. Combinatorial designs can be thought of as hypergraphs with useful and interesting characteristics. In this paper we explore the burning and lazy burning processes in Steiner triple systems (STSs). We obtain logarithmic bounds on the burning number of an arbitrary STS and prove the existence of an STS with burning number $\rho$ for any integer $\rho \geq 3$. We observe that the concept of lazy burning is equivalent to the notion of the dimension of an STS, and we consider ramifications of this equivalence. We also show that the difference between the burning and lazy burning number of a Steiner triple system can be arbitrarily large. Finally, we consider burning and lazy burning numbers of affine and projective triple systems in detail.

math.CO↗

arXiv2026

Locally Semi-Equitable Colourings of BIBDs

We study $\ell$-colourings of $(v,k,λ)$-BIBDs (balanced incomplete block designs) where, within each block, one colour is absent and each of the $\ell-1$ other colours appears exactly $\frac{k}{\ell-1}$ times. We establish several necessary conditions for such colourings to exist. We also use these coloured BIBDs to provide new necessary conditions for the existence of Hadamard matrices, affine planes, and twin prime powers.

math.CO↗

arXiv2026

Uniquely 2-colourable 4-cycle decompositions

A cycle system of order $n$ is a decomposition of the edges of the complete graph $K_n$ into cycles of a fixed length. A cycle system is said to be $k$-colourable if we can assign $k$ colours to its vertices so that no cycle is monochromatic. A $k$-colourable cycle system is uniquely $k$-colourable if its colouring is unique up to the permutation of colour classes. In this paper, we construct uniquely $2$-colourable $4$-cycle systems of order $n$ for all admissible $n\geq 49$, and also uniquely $2$-colourable $4$-cycle decompositions of $K_n - I$, for all admissible $n \geq 50$. These constructions contribute to the broader study of uniquely colourable cycle systems and open new directions for future research.

math.CO↗

arXiv2026

Colourings of Uniform Group Divisible Designs and Maximum Packings

A weak $c$-colouring of a design is an assignment of colours to its points from a set of $c$ available colours, such that there are no monochromatic blocks. A colouring of a design is block-equitable, if for each block, the number of points coloured with any available pair of colours differ by at most one. Weak and block-equitable colourings of balanced incomplete block designs have been previously considered. In this paper, we extend these concepts to group divisible designs (GDDs) and packing designs. We first determine when a $k$-GDD of type $g^u$ can have a block-equitable $c$-colouring. We then give a direct construction of maximum block-equitable $2$-colourable packings with block size $4$; a recursive construction has previously appeared in the literature. We also generalise a bound given in the literature for the maximum size of block-equitably $2$-colourable packings to $c>2$. Furthermore, we establish the asymptotic existence of uniform $k$-GDDs with arbitrarily many groups and arbitrary chromatic numbers (with the exception of $c=2$ and $k=3$). A structural analysis of $2$- and $3$-uniform $3$-GDDs obtained from 4-chromatic STS$(v)$ where $v\in\{21,25,27,33,37,39\}$ is given. We briefly discuss weak colourings of packings, and finish by considering some further constraints on weak colourings of GDDs, namely requiring all groups to be either monochromatic or equitably coloured.

math.CO↗

arXiv2025

Extending Graph Burning to Hypergraphs

Graph burning is a round-based game or process that discretely models the spread of influence throughout a network. We introduce a generalization of graph burning which applies to hypergraphs, as well as a variant called ''lazy'' hypergraph burning. Interestingly, lazily burning a graph is trivial, while lazily burning a hypergraph can be quite complicated. Moreover, the lazy burning model is a useful tool for analyzing the round-based model. One of our key results is that arbitrary hypergraphs do not satisfy a bound analogous to the one in the Burning Number Conjecture for graphs. We also obtain bounds on the burning number and lazy burning number of a hypergraph in terms of its parameters, and present several open problems in the field of (lazy) hypergraph burning.

math.CO↗

arXiv2025

Distance-Restricted Firefighting on Finite Graphs

In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph $G$ and then $b$ firefighters protect $b$ vertices. On each subsequent turn, the fire spreads to the collective unburned neighbourhood of all the burning vertices and the firefighters again protect $b$ vertices. Once a vertex has been burned or protected it remains that way for the rest of the game. In \textit{distance-restricted firefighting} the firefighters' movement is restricted so they can only move up to some fixed distance $d$ and they may or may not be permitted to move through burning vertices. In this paper we establish the NP-completeness of the distance-restricted versions of {\sc $b$-Firefighter} and present an integer program for computing the exact value. We also discuss some interesting properties of the \textit{Expected Damage} function.

math.CO↗

arXiv2025

Weak colourings of Kirkman triple systems

A $δ$-colouring of the point set of a block design is said to be {\em weak} if no block is monochromatic. The {\em chromatic number} $χ(S)$ of a block design $S$ is the smallest integer $δ$ such that $S$ has a weak $δ$-colouring. It has previously been shown that any Steiner triple system has chromatic number at least $3$ and that for each $v\equiv 1$ or $3\pmod{6}$ there exists a Steiner triple system on $v$ points that has chromatic number $3$. Moreover, for each integer $δ\geq 3$ there exist infinitely many Steiner triple systems with chromatic number $δ$. We consider colourings of the subclass of Steiner triple systems which are resolvable. We show that for each $v\equiv 3\pmod{6}$ there exists a Kirkman triple system on $v$ points with chromatic number $3$. We also show that for each integer $δ\geq 3$, there exist infinitely many Kirkman triple systems with chromatic number $δ$. We close with several open problems.

math.CO↗

arXiv2024

Brushing Directed Graphs

Brushing of graphs is a graph searching process in which the searching agents are called brushes. We focus on brushing directed graphs based on a new model in which the brushes can only travel in the same direction as the orientation of the arcs that they traverse. We discuss strategies to brush directed graphs as well as values and bounds for the brushing number of directed graphs. We determine the brushing number for any transitive tournament, which we use to give an upper bound for the brushing number of directed acyclic graphs in general. We also establish exact values for the brushing numbers of complete directed graphs, rooted trees, and rotational tournaments.

math.CO↗

arXiv2024

Proportion-Based Hypergraph Burning

Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to hypergraphs. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads - if some fixed proportion of vertices are on fire in a hyperedge, then in the next round the entire hyperedge catches fire. This new variant has more potential for applications than the original model, and it is similarly viable for obtaining deep theoretical results. We obtain bounds which apply to general hypergraphs, and introduce the concept of the burning distribution, which describes how the model changes as the proportion ranges over (0,1). We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.

math.CO↗

arXiv2024

Existential Closure in Uniform Hypergraphs

For a positive integer $n$, a graph with at least $n$ vertices is $n$-existentially closed or simply $n$-e.c. if for any set of vertices $S$ of size $n$ and any set $T\subseteq S$, there is a vertex $x\not\in S$ adjacent to each vertex of $T$ and no vertex of $S\setminus T$. We extend this concept to uniform hypergraphs, find necessary conditions for $n$-e.c. hypergraphs to exist, and prove that random uniform hypergraphs are asymptotically $n$-existentially closed. We then provide constructions to generate infinitely many examples of $n$-e.c. hypergraphs. In particular, these constructions use certain combinatorial designs as ingredients, adding to the ever-growing list of applications of designs.

math.CO↗

arXiv2023

The Edge-Connectivity of Vertex-Transitive Hypergraphs

A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise this result to hypergraphs and show that every connected linear uniform vertex-transitive hypergraph is maximally edge-connected. We also show that if we relax either the linear or uniform conditions in this generalisation, then we can construct examples of vertex-transitive hypergraphs which are not maximally edge-connected.

math.CO↗

arXiv2023

Existential Closure in Line Graphs

A graph $G$ is {\it $n$-existentially closed} if, for all disjoint sets of vertices $A$ and $B$ with $|A\cup B|=n$, there is a vertex $z$ not in $A\cup B$ adjacent to each vertex of $A$ and to no vertex of $B$. In this paper, we investigate $n$-existentially closed line graphs. In particular, we present necessary conditions for the existence of such graphs as well as constructions for finding infinite families of such graphs. We also prove that there are exactly two $2$-existentially closed planar line graphs. We then consider the existential closure of the line graphs of hypergraphs and present constructions for $2$-existentially closed line graphs of hypergraphs.

math.CO↗

arXiv2022

Colourings of path systems

A $P_m$ path in a graph is a path on $m$ vertices. A $P_m$ system of order $n>1$ is a partition of the edges of the complete graph $K_n$ into $P_m$ paths. A $P_m$ system is said to be $k$-colourable if the vertex set of $K_n$ can be partitioned into $k$ sets called colour classes such that no path in the system is monochromatic. The system is $k$-chromatic if it is $k$-colourable but is not $(k-1)$-colourable. If every $k$-colouring of a $P_m$ system can be obtained from some $k$-colouring $ϕ$ by a permutation of the colours, we say that the system is uniquely $k$-colourable. In this paper, we first observe that there exists a $k$-chromatic $P_m$ system for any $k\geq 2$ and $m\geq 4$ where $m$ is even. Next, we prove that there exists an equitably 2-chromatic $P_4$ system of order $n$ for each admissible order $n$. We then show that for all $k\geq 3$, there exists a $k$-chromatic $P_4$ system of order $n$ for all sufficiently large admissible $n$. Finally, we show that there exists a uniquely 2-chromatic $P_4$ system of order $n$ for each admissible $n \geq 109$.

math.CO↗

arXiv2022

Mutually orthogonal cycle systems

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $Γ$ is a set of ${\ell}$-cycles which partition the edge set of $Γ$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct cycles from ${\mathcal F}\cup {\mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $\ell$-cycle systems of $Γ$ is said to be a set of mutually orthogonal cycle systems of $Γ$. Let $μ(\ell,n)$ (respectively, $μ'(\ell,n)$) be the maximum integer $μ$ such that there exists a set of $μ$ mutually orthogonal (cyclic) $\ell$-cycle systems of the complete graph $K_n$. We show that if $\ell\geq 4$ is even and $n\equiv 1\pmod{2\ell}$, then $μ'(\ell,n)$, and hence $μ(\ell,n)$, is bounded below by a constant multiple of $n/\ell^2$. In contrast, we obtain the following upper bounds: $μ(\ell,n)\leq n-2$; $μ(\ell,n)\leq (n-2)(n-3)/(2(\ell-3))$ when $\ell \geq 4$; $μ(\ell,n)\leq 1$ when $\ell>n/\sqrt{2}$; and $μ'(\ell,n)\leq n-3$ when $n \geq 4$. We also obtain computational results for small values of $n$ and $\ell$.

math.CO↗

arXiv2022

Acyclic polynomials of graphs

For each nonnegative integer $i$, let $a_i$ be the number of $i$-subsets of $V(G)$ that induce an acyclic subgraph of a given graph $G$. We define $A(G,x) = \sum_{i \geq 0} a_i x^i$ (the generating function for $a_i$) to be the acyclic polynomial for $G$. After presenting some properties of these polynomials, we investigate the nature and location of their roots.

math.CO↗

arXiv2021

Cops that surround a robber

We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber's neighbouring vertices. We denote by $σ(G)$ the {\em surrounding cop number} of $G$, namely the least number of cops required to surround a robber in the graph $G$. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest include product graphs, graphs arising from combinatorial designs, and generalised Petersen graphs.

math.CO↗

arXiv2020

The Firebreak Problem

Suppose we have a network that is represented by a graph $G$. Potentially a fire (or other type of contagion) might erupt at some vertex of $G$. We are able to respond to this outbreak by establishing a firebreak at $k$ other vertices of $G$, so that the fire cannot pass through these fortified vertices. The question that now arises is which $k$ vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the $k$ vertices of the firebreak. This is the essence of the {\sc Firebreak} decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant-bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems.

math.CO↗

arXiv2019

A Perfect One-Factorisation of $K_{56}$

In 1963, Anton Kotzig conjectured that for each $n \geq 2$ the complete graph $K_{2n}$ has a perfect one-factorisation (i.e., a decomposition into perfect matchings such that each pair of perfect matchings of the decomposition induces a Hamilton cycle). We affirmatively settle the smallest unresolved case for this conjecture.

math.CO↗