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papers

Publications (12)

math.CO2018

Stability results for graphs with a critical edge

Alexander Roberts, Alex Scott

The classical stability theorem of ErdÅs and Simonovits states that, for any fixed graph with chromatic number $k + 1 \geq 3$ , the following holds: every $n$ -vertex graph that is $H$ …

math.CO2016

Tree Matchings

Alexander Roberts

An $(s, t)$ -matching in a bipartite graph $G = (U, V, E)$ is a subset of the edges $F$ such that each component of $G [F]$ is a tree with at most $t$ edges and each vertex in $U$ has $s$ …

math.CO2019

Shotgun reconstruction in the hypercube

MichaÅ Przykucki, Alexander Roberts, Alex Scott

Mossel and Ross raised the question of when a random colouring of a graph can be reconstructed from local information, namely the colourings (with multiplicity) of balls of given r…

math.PR2021

Parking on the integers

MichaÅ Przykucki, Alexander Roberts, Alex Scott

Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Da…

math.CO2023

Decomposing random permutations into order-isomorphic subpermutations

Carla Groenland, Tom Johnston, DÃ¡niel KorÃ¡ndi +3

Two permutations $s$ and $t$ are $k$ -similar if they can be decomposed into subpermutations $s^{1}, \dots, s^{k}$ and $t^{1}, \dots, t^{k}$ such that $s^{i}$ is order-isomorphic to $t^{i}$ f…

math.CO2019

Vertex-isoperimetric stability in the hypercube

MichaÅ Przykucki, Alexander Roberts

Harper's Theorem states that, in a hypercube, among all sets of a given fixed size the Hamming balls have minimal closed neighbourhoods. In this paper we prove a stability-like res…

cond-mat.mes-hall2025

Chirality Imprinting and Spin-texture Tunability in Conformally Coated 3D Magnetic Nanostructured Metamaterials

Alexander Roberts, Huixin Guo, Joseph Askey +4

Three-dimensional (3D) magnetic nanostructures offer unprecedented opportunities for engineering emergent spin textures, but controlling their configuration remains a central chall…

math.CO2021

Exact stability for TurÃ¡n's Theorem

DÃ¡niel KorÃ¡ndi, Alexander Roberts, Alex Scott

TurÃ¡n's Theorem says that an extremal $K_{r + 1}$ -free graph is $r$ -partite. The Stability Theorem of ErdÅs and Simonovits shows that if a $K_{r + 1}$ -free graph with $n$ vertices ha…

math.CO2018

Approximating the position of a hidden agent in a graph

Hannah Guggiari, Alexander Roberts, Alex Scott

A cat and mouse play a pursuit and evasion game on a connected graph $G$ with $n$ vertices. The mouse moves to vertices $m_{1}, m_{2}, \dots$ of $G$ where $m_{i}$ is in the closed neighbou…

math.CO2020

Maximising the Number of Cycles in Graphs with Forbidden Subgraphs

Natasha Morrison, Alexander Roberts, Alex Scott

Fix $k \geq 2$ and let $H$ be a graph with $I ¨ (H) = k + 1$ containing a critical edge. We show that for sufficiently large $n$ , the unique $n$ -vertex $H$ -free graph containing the max…

math.CO2025

Shotgun assembly of random graphs

Tom Johnston, Gal Kronenberg, Alexander Roberts +1

In the graph shotgun assembly problem, we are given the balls of radius $r$ around each vertex of a graph and asked to reconstruct the graph. We study the shotgun assembly of the E…

math.CO2020

Isoperimetric stability in lattices

Ben Barber, Joshua Erde, Peter Keevash +1

We obtain isoperimetric stability theorems for general Cayley digraphs on $Z^{d}$ . For any fixed $B$ that generates $Z^{d}$ over $Z$ , we characterise the app…