Alexander Clow

activity

20212026

most citedOriented Colouring Graphs of Bounded Degree and Degeneracy

1 citations · 1 across the 13 of their papers we have counts for

collaborators

18 papers

math.CO2026

MAD Phase Transitions in the Oriented Chromatic Number

Alexander Clow

For an oriented graph $G$ the oriented chromatic number of $G$ , written $χ_{o} (G)$ , is the least integer $t$ such that $G$ has a homomorphism to a tournament on $t$ vertices. The ori…

math.CO2026

Greedily Constructing Small Quasi-Kernels

Alexander Clow

In a digraph $D$ ,a quasi-kernel is an independent set $Q$ such that for every vertex $u$ , there is a vertex $v \in Q$ satisfying $dist (v, u) \leq 2$ . In 1974 Chvátal and Lovás…

math.CO2025

Improved Bounds for the Ultimate Independence Ratio of Odd Wheels

Alexander Clow, Hitesh Kumar, Shivaramakrishna Pragada

The ultimate independence ratio of a graph $G$ is defined as $I (G) = lim_{k \to \infty} \frac{α ( G ^{□ k} )}{∣ V ( G ) ∣ ^{k}},$ where $α (G^{□ k})$ is the independence…

math.CO2025

A Note on Large Degenerate Induced Subgraphs in Sparse Graphs

Alexander Clow, Sean Kim, Ladislav Stacho

Given a graph $G$ and a non-negative integer $d$ let $α_{d} (G)$ be the order of a largest induced $d$ -degenerate subgraph of $G$ . We prove that for any pair of non-negative integers…

math.CO2025

Cops and Robbers on Graphs with Path Constraints

Alexander Clow, Erin Meger

In 2019, Sivaraman conjectured that every $P_{k}$ -free graph has cop number at most $k - 3$ . In the same year, Liu proved this conjecture for $(P_{k}, claw)$ -free graphs. Recently…

math.CO2025

Tight Bounds for Cycle-Edge Decompositions and Covers

Saieed Akbari, Jonny Aloni, Arash Beikmohammadi +1

An old conjecture of Erd{ő}s and Gallai states that every $n$ vertex graph can be decomposed, that is $E (G)$ can be partitioned, into $O (n)$ cycles and edges. The covering version…

math.CO2025

Cops and Robbers, Clique Covers, and Induced Cycles

Alexander Clow, Imed Zaguia

We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$ , the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(…

math.CO2025

On Gyárfás' Path-Colour Problem

Ben Cameron, Alexander Clow

In their 1997 paper titled ``Fruit Salad", Gyárfás posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$ -colourable subgraph, th…

math.CO2025

A Counterexample to a Conjecture of Lovász

Alexander Clow, Penny Haxell, Bojan Mohar

In 1975 Lovász conjectured that every $r$ -partite, $r$ -uniform hypergraph contains $r - 1$ vertices whose deletion reduces the matching number. If true, this statement would imply a…

math.CO2024

On Oriented Colourings of Graphs on Surfaces

Alexander Clow

For an oriented graph $G$ , the least number of colours required to oriented colour $G$ is called the oriented chromatic number of $G$ and denoted $χ_{o} (G)$ .For a non-negative intege…

cs.FL2024

A cornering strategy for synchronizing a DFA

Peter Bradshaw, Alexander Clow, Ladislav Stacho

This paper considers the existence of short synchronizing words in deterministic finite automata (DFAs). We define two general strategies for generating synchronizing words, and we…

math.CO2023

Expanders Satisfy the Weak Meyniel Conjecture

Alexander Clow

We show that if ${G_{n}}_{n \geq 1}$ is a sequence of graphs of order $n$ with bounded maximum degree and isoperimetric function $Φ (G_{n}, n^{1 - α})$ bounded away from $0$ as $n\rightar…

math.CO2023

Injective edge colorings of degenerate graphs and the oriented chromatic number

Peter Bradshaw, Alexander Clow, Jingwei Xu

Given a graph $G$ , an injective edge-coloring of $G$ is a function $ψ : E (G) \to N$ such that if $ψ (e) = ψ (e^{'})$ , then no third edge joins an endpoint of $e$ and an en…

math.CO2023

Eternal Distance-2 Domination in Trees

Alexander Clow, Christopher M van Bommel

We consider the eternal distance-2 domination problem, recently proposed by Cox, Meger, and Messinger, on trees. We show that finding a minimum eternal distance-2 dominating set of…

math.CO2023

Graphs with Large Girth and Small Cop Number

Alexander Clow

In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when $G$ is graph of girth $g$ and the minimum degree $δ \geq 2$ , then $c(G) = O(n\log…

math.CO2023

Ordinal Sums of Numbers

Alexander Clow, Neil McKay

In this paper we consider ordinal sums of combinatorial games where each summand is a number, not necessarily in canonical form. In doing so we give formulas for the value of an or…

math.CO2023★ 1 cited

Oriented Colouring Graphs of Bounded Degree and Degeneracy

Alexander Clow, Ladislav Stacho

This paper considers upper bounds on the oriented chromatic number $χ_{o} (G)$ , of an oriented graph $G$ in terms of its $2$ -dipath chromatic number $χ_{2} (G)$ , degeneracy $d (G)$ , and m…

math.CO2021

Advances in finding ideal play on poset games

Alexander Clow, Stephen Finbow

Poset games are a class of combinatorial game that remain unsolved. Soltys and Wilson proved that computing wining strategies is in \textbf{PSPACE} and aside from special cases suc…