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The structure of FAC posets and the Aharoni--Korman conjecture
Lawrence Hollom
A poset is said to satisfy the finite antichain condition, or FAC for short, if it has no infinite antichain. Such posets exhibit rich and complex structure, and it was conject…
Hypercube geodesics with few colour changes
Lawrence Hollom
What is the maximum, over all 2-colourings of the edges of the -dimensional hypercube , of the minimal number of times a path between a vertex and its antipode $\bar{v}…
Uniformly balanced -factors in multicoloured complete graphs
Agnijo Banerjee, Lawrence Hollom
A balanced colouring of a graph is one in which every colour appears the same number of times. Given a fixed graph on vertices and a balanced -colouring of the complete…
Counterexamples to conjectures on strong maximality and minimality
Lawrence Hollom, Benedict Randall Shaw
We provide counterexamples to several conjectures concerning strongly maximal and strongly minimal structures in infinite graphs and hypergraphs. In particular, we construct 3-unif…
Reverse Littlewood--Offord problems with parity conditions
Lawrence Hollom, Gregory B. Sorkin
We consider the probability that the random signed sum lies within a given distance of the origin, where are fixe…
Connecting hypercube 1-factors
Lawrence Hollom, Benedict Randall Shaw
A 1-factorisation of a regular graph is a partition of its edge set into perfect matchings of . Behague asked for the minimal such that some -factorisatio…
Approximate Itai-Zehavi conjecture for random graphs
Lawrence Hollom, Lyuben Lichev, Adva Mond +2
A famous conjecture by Itai and Zehavi states that, for every -vertex-connected graph and every vertex in , there are spanning trees of such that, for every v…
Finding long cycles in a percolated expander graphs
Lawrence Hollom
Given a graph , the percolated graph has each edge independently retained with probability . Collares, Diskin, Erde, and Krivelevich initiated the study of large struct…
Monotonicity and decompositions of random regular graphs
Lawrence Hollom, Lyuben Lichev, Adva Mond +2
In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameter…
Double-jump phase transition for the reverse Littlewood--Offord problem
Lawrence Hollom, Julien Portier, Victor Souza
Erdős conjectured in 1945 that for any unit vectors in and signs taken independently and uniformly in $\{-1…
A note on high-dimensional discrepancy of subtrees
Lawrence Hollom, Lyuben Lichev, Adva Mond +1
For a tree and a function , the imbalance of a subtree is given by . The -dimensional discrepancy o…
The Aharoni--Korman conjecture is false
Lawrence Hollom
A poset is said to satisfy the finite antichain condition, or FAC, if it has no infinite antichain. It was conjectured by Aharoni and Korman in 1992 that any FAC poset poss…
Discrepancies of spanning trees in dense graphs
Lawrence Hollom, Lyuben Lichev, Adva Mond +1
We address several related problems on combinatorial discrepancy of trees in a setting introduced by Erdős, Füredi, Loebl and Sós. Given a fixed tree on vertices and an edg…
A uniform bound on almost colour-balanced perfect matchings in colour-balanced cliques
Lawrence Hollom
An edge-colouring of a graph is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matc…
Almost colour-balanced spanning forests in complete graphs
Lawrence Hollom, Adva Mond, Julien Portier
Given whose edges are coloured red and blue, and a forest of order , we seek embeddings of with small imbalance, that is, difference between the numbers of red and…
The bunkbed conjecture is not robust to generalisation
Lawrence Hollom
The bunkbed conjecture, which has featured in the folklore of probability theory since at least 1985, concerns bond percolation on the product graph . We have two copies…
On conjectures concerning the graph grabbing game
Lawrence Hollom
We consider two conjectures made in regard to the graph grabbing game, played on a vertex weighted graph. Seacrest and Seacrest conjectured in 2012 that the first player can win th…
On graphs with maximum difference between game chromatic number and chromatic number
Lawrence Hollom
In the vertex colouring game on a graph , Maker and Breaker alternately colour vertices of from a palette of colours, with no two adjacent vertices allowed the same colo…
A note on monotonicity in Maker-Breaker graph colouring games
Lawrence Hollom
In the Maker-Breaker vertex colouring game, first publicised by Gardner in 1981, Maker and Breaker alternately colour vertices of a graph using a fixed palette, maintaining a prope…
Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem
Lawrence Hollom, Julien Portier
In this paper we prove that for every finite Rademacher sum , confirming a conjecture by Hitczenko and Kwapie{ń} from 1994,…
On interval colourings of graphs
Lawrence Hollom, Julien Portier, Leo Versteegen
An interval colouring of a graph is a proper colouring such that the set of colours of edges incident to any given vertex forms an interval of $…
A note on interval colourings of graphs
Maria Axenovich, António Girão, Lawrence Hollom +5
A graph is said to be interval colourable if it admits a proper edge-colouring using palette in which the set of colours incident to each vertex is an interval. The in…
A new proof of the bunkbed conjecture in the limit
Lawrence Hollom
For a finite simple graph , the bunkbed graph is defined to be the product graph . We will label the two copies of a vertex as and .…