arXiv subjects

Lawrence Hollom

Publications and source records attributed to Lawrence Hollom.

23 records · Page 2Linked to original sources

arXiv2023

On conjectures concerning the graph grabbing game

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 the graph grabbing game on any even-order bipartite graph. Eoh and Choi conjectured a strengthening of this in 2019, namely that the first player can win on any graph with no induced corona product of an odd cycle and a point. We provide a family of counterexamples to the latter conjecture, and propose a weaker conjecture in its place. We also show that the above two conjectures are equivalent when the vertex weights are all $0$ or $1$.

math.CO↗

arXiv2023

Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem

In this paper we prove that $\mathbb{P}(|X| \geq \sqrt{\text{Var}(X)}) \geq 7/32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{ń} from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvořák and Klein. Moreover we fully determine the function $f(y)= \inf_X \mathbb{P}(|X| \geq y\sqrt{\text{Var}(X)})$ where the $\inf$ is taken over all finite Rademacher sums $X$, confirming a conjecture by Lowther and giving a partial answer to a question by Keller and Klein.

math.CO↗

arXiv2023

On interval colourings of graphs

An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\colon E\to \mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\mathbb{Z}$. The interval thickness $θ(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $θ(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \frac{\log n}{\log \log n} \leq θ(n) \leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan. In the second part of the paper, we confirm a conjecture of Axenovich that the maximum number of colours used in an interval colouring of a planar graph on $n$ vertices is at most $3n/2-2$.

math.CO↗

arXiv2023

A note on interval colourings of graphs

A graph is said to be interval colourable if it admits a proper edge-colouring using palette $\mathbb{N}$ in which the set of colours incident to each vertex is an interval. The interval colouring thickness of a graph $G$ is the minimum $k$ such that $G$ can be edge-decomposed into $k$ interval colourable graphs. We show that $θ(n)$, the maximum interval colouring thickness of an $n$-vertex graph, satisfies $θ(n) =Ω(\log(n)/\log\log(n))$ and $θ(n)\leq n^{5/6+o(1)}$, which improves on the trivial lower bound and an upper bound of the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an $n$-vertex planar graph uses at most $3n/2-2$ colours.

math.CO↗

arXiv2023

A new proof of the bunkbed conjecture in the $p\uparrow 1$ limit

For a finite simple graph $G$, the bunkbed graph $G^\pm$ is defined to be the product graph $G\square K_2$. We will label the two copies of a vertex $v\in V(G)$ as $v_-$ and $v_+$. The bunkbed conjecture, posed by Kasteleyn, states that for independent bond percolation on $G^\pm$, percolation from $u_-$ to $v_-$ is at least as likely as percolation from $u_-$ to $v_+$, for any $u,v\in V(G)$. Despite the plausibility of this conjecture, so far the problem in full generality remains open. Recently, Hutchcroft, Nizić-Nikolac, and Kent gave a proof of the conjecture in the $p\uparrow 1$ limit. Here we present a new proof of the bunkbed conjecture in this limit, working in the more general setting of allowing different probabilities on different edges of $G^\pm$.

math.CO↗