Richard Mycroft

activity

20082026

most citedHamilton l-cycles in uniform hypergraphs

81 citations · 322 across the 17 of their papers we have counts for

collaborators

27 papers

math.CO2026

Towards Pósa's Conjecture for $3$ -graphs

Debmalya Bandyopadhyay, Allan Lo, Richard Mycroft

We prove that every $3$ -graph $H$ on $n$ vertices with minimum codegree $δ_{2} (H) \geq 7 n /9 + o (n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Beden…

math.CO2025

Two new results on maximal left-compressed intersecting families

Allan Flower, Richard Mycroft

This paper presents two new results on the theory of maximal left-compressed intersecting families (MLCIFs). First, we answer a question raised by Barber by showing that the number…

math.CO2025

Positive codegree thresholds for perfect matchings in hypergraphs

Richard Mycroft, Camila Zárate-Guerén

We give, for each $k \geq 3$ , the precise best possible minimum positive codegree condition for a perfect matching in a large $k$ -uniform hypergraph $H$ on $n$ vertices. Specifical…

math.CO2025

Positive codegree thresholds for Hamilton cycles in hypergraphs

Richard Mycroft, Camila Zárate-Guerén

For each $k \geq 3$ and $1 \leq ℓ \leq k - 1$ we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton $ℓ$ -cycle in a $k$ -uni…

math.CO2025

Perfect tilings of 3-graphs with the generalised triangle

Candida Bowtell, Amarja Kathapurkar, Natasha Morrison +1

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$ -uniform hypergraph $H$ with copies of the generalised triangle $T$ , which is…

math.CO2024

Matchings in multipartite hypergraphs

Candida Bowtell, Richard Mycroft

A folklore result on matchings in graphs states that if $G$ is a bipartite graph whose vertex classes $A$ and $B$ each have size $n$ , with $deg (u) \geq a$ for every $u \in…

math.CO2023

Removing induced powers of cycles from a graph via fewest edits

Amarja Kathapurkar, Richard Mycroft

What is the minimum proportion of edges which must be added to or removed from a graph of density $p$ to eliminate all induced cycles of length $h$ ? The maximum of this quantity ov…

math.CO2020

Trees and treelike structures in dense digraphs

Richard Mycroft, Tássio Naia

We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at le…

math.CO2018

Classification of maximum hittings by large families

Candida Bowtell, Richard Mycroft

For integers $r$ and $n$ , where $n$ is sufficiently large, and for every set $X \subseteq [n]$ we determine the maximal left-compressed intersecting families $\mathcal{A}\subseteq…

math.CO2018

Hamilton $ℓ$ -cycles in randomly-perturbed hypergraphs

Andrew McDowell, Richard Mycroft

We prove that for integers $2 \leq ℓ < k$ and a small constant $c$ , if a $k$ -uniform hypergraph with linear minimum codegree is randomly `perturbed' by changing non-edges to edg…

math.CO2016

Contagious sets in a degree-proportional bootstrap percolation process

Frederik Garbe, Andrew McDowell, Richard Mycroft

We study the following bootstrap percolation process: given a connected graph $G$ , a constant $ρ \in [0, 1]$ and an initial set $A \subseteq V (G)$ of \emph{infected} vertices, at ea…

math.CO2016

Unavoidable trees in tournaments

Richard Mycroft, Tássio Naia

An oriented tree $T$ on $n$ vertices is unavoidable if every tournament on $n$ vertices contains a copy of $T$ . In this paper we give a sufficient condition for $T$ to be unavoidab…

math.CO2016

Hamilton cycles in hypergraphs below the Dirac threshold

Frederik Garbe, Richard Mycroft

We establish a precise characterisation of $4$ -uniform hypergraphs with minimum codegree close to $n /2$ which contain a Hamilton $2$ -cycle. As an immediate corollary we identify th…

math.CO2016

Triangle-tilings in graphs without large independent sets

József Balogh, Andrew McDowell, Theodore Molla +1

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an $n$ -vertex graph $G$ with sublinear independence number. In th…

math.CO2016

The minimum vertex degree for an almost-spanning tight cycle in a $3$ -uniform hypergraph

Oliver Cooley, Richard Mycroft

We prove that any $3$ -uniform hypergraph whose minimum vertex degree is at least $(\frac{5}{9} + o (1)) (2 n)$ admits an almost-spanning tight cycle, that is, a ti…

math.CO2016★ 3 cited

An asymptotic multipartite Kühn-Osthus theorem

Ryan R. Martin, Richard Mycroft, Jozef Skokan

In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph $H$ with chromatic number $r$ , the asymptotic mu…

math.CO2015

Hamilton cycles in quasirandom hypergraphs

John Lenz, Dhruv Mubayi, Richard Mycroft

We show that, for a natural notion of quasirandomness in $k$ -uniform hypergraphs, any quasirandom $k$ -uniform hypergraph on $n$ vertices with constant edge density and minimum vert…

math.CO2014

Tight cycles and regular slices in dense hypergraphs

Peter Allen, Julia Böttcher, Oliver Cooley +1

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these…

math.CO2014★ 24 cited

A random version of Sperner's theorem

József Balogh, Richard Mycroft, Andrew Treglown

Let $P (n)$ denote the power set of $[n]$ , ordered by inclusion, and let $P (n, p)$ be obtained from $P (n)$ by selecting elements from $P (n)$ i…

math.CO2014★ 2 cited

Packing k-partite k-uniform hypergraphs

Richard Mycroft

Let $G$ and $H$ be $k$ -graphs ( $k$ -uniform hypergraphs); then a perfect $H$ -packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex…

math.CO2013★ 29 cited

Polynomial-time perfect matchings in dense hypergraphs

Peter Keevash, Fiachra Knox, Richard Mycroft

Let $H$ be a $k$ -graph on $n$ vertices, with minimum codegree at least $n / k + c n$ for some fixed $c > 0$ . In this paper we construct a polynomial-time algorithm which finds either…

math.CO2012★ 43 cited

A Multipartite Hajnal-Szemerédi Theorem

Peter Keevash, Richard Mycroft

The celebrated Hajnal-Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analog…

math.CO2011

A Geometric Theory for Hypergraph Matching

Peter Keevash, Richard Mycroft

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric,…

math.CO2010★ 35 cited

A proof of Sumner's universal tournament conjecture for large tournaments

Daniela Kühn, Richard Mycroft, Deryk Osthus

Sumner's universal tournament conjecture states that any tournament on $2 n - 2$ vertices contains any directed tree on $n$ vertices. In this paper we prove that this conjecture holds…

Towards Pósa's Conjecture for 3-graphs

Two new results on maximal left-compressed intersecting families

Positive codegree thresholds for perfect matchings in hypergraphs

Positive codegree thresholds for Hamilton cycles in hypergraphs

Perfect tilings of 3-graphs with the generalised triangle

Matchings in multipartite hypergraphs

Removing induced powers of cycles from a graph via fewest edits

Trees and treelike structures in dense digraphs

Classification of maximum hittings by large families

Hamilton ℓ-cycles in randomly-perturbed hypergraphs

Contagious sets in a degree-proportional bootstrap percolation process

Unavoidable trees in tournaments

Hamilton cycles in hypergraphs below the Dirac threshold

Triangle-tilings in graphs without large independent sets

The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

An asymptotic multipartite Kühn-Osthus theorem

Hamilton cycles in quasirandom hypergraphs

Tight cycles and regular slices in dense hypergraphs

A random version of Sperner's theorem

Packing k-partite k-uniform hypergraphs

Polynomial-time perfect matchings in dense hypergraphs

A Multipartite Hajnal-Szemerédi Theorem

A Geometric Theory for Hypergraph Matching

A proof of Sumner's universal tournament conjecture for large tournaments

Towards Pósa's Conjecture for $3$ -graphs

Hamilton $ℓ$ -cycles in randomly-perturbed hypergraphs

The minimum vertex degree for an almost-spanning tight cycle in a $3$ -uniform hypergraph