arXiv subjects

John Haslegrave

Publications and source records attributed to John Haslegrave.

50 records · Page 3Linked to original sources

arXiv2018

Spanning surfaces in 3-graphs

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface $\mathscr{S}$, we show that any two-dimensional simplicial complex on $n$ vertices in which each pair of vertices belongs to at least $n/3 + o(n)$ facets contains a homeomorph of $\mathscr{S}$ spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on $n$ vertices with minimum codegree exceeding $n/3+o(n)$ contains a spanning triangulation of the $2$-sphere.

math.CO↗

arXiv2018

Non-convergence of proportions of types in a preferential attachment graph with three co-existing types

We consider the preferential attachment model with multiple vertex types introduced by Antunović, Mossel and Rácz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunović, Mossel and Rácz. We also consider another family of examples where we show that the conjecture does hold.

math.PR↗

arXiv2018

Proof of a local antimagic conjecture

An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Bača \& Semaničová-Feňovčíková (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .

math.CO↗

arXiv2017

Locating a robber with multiple probes

We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any $n$-vertex graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$ obtained by replacing each edge of $G$ by a path of length $m$, if $m\geq n$. The present authors showed that, for all but a few small values of $n$, this bound may be improved to $m\geq n/2$, which is best possible. In this paper we consider the natural extension in which the cop probes a set of $k$ vertices, rather than a single vertex, at each turn. We consider the relationship between the value of $k$ required to ensure victory on the original graph and the length of subdivisions required to ensure victory with $k=1$. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of $k$ for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree $Δ$, which is best possible up to a factor of $(1-o(1))$.

math.CO↗

arXiv2017

An evasion game on a graph

This paper introduced a pursuit and evasion game to be played on a connected graph. One player moves invisibly around the graph, and the other player must guess his position. At each time step the second player guesses a vertex, winning if it is the current location of the first player; if not the first player must move along an edge. It is shown that the graphs on which the second player can guarantee to win are precisely the trees that do not contain a particular forbidden subgraph, and best possible capture times on such graphs are obtained.

math.CO↗

arXiv2017

Judicious partitions of uniform hypergraphs

The vertices of any graph with $m$ edges may be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollobás and Thomason conjectured that the vertices of any $r$-uniform hypergraph with $m$ edges may likewise be partitioned into $r$ classes such that each part meets at least $\frac{r}{2r-1}m$ edges. In this paper we prove the weaker statement that, for each $r\ge 4$, a partition into $r$ classes may be found in which each class meets at least $\frac{r}{3r-4}m$ edges, a substantial improvement on previous bounds.

math.CO↗

arXiv2016

Reaching consensus on a connected graph

We study a simple random process in which vertices of a connected graph reach consensus through pairwise interactions. We compute outcome probabilities, which do not depend on the graph structure, and consider the expected time until a consensus is reached. In some cases we are able to show that this is minimised by $K_n$. We prove an upper bound for the case $p=0$ and give a family of graphs which asymptotically achieve this bound. In order to obtain the mean of the waiting time we also study a gambler's ruin process with delays. We give the mean absorption time and prove that it monotonically increases with $p\in[0,1/2]$ for symmetric delays.

math.PR↗

arXiv2016

Majority dynamics with one nonconformist

We consider a system in which a group of agents represented by the vertices of a graph synchronously update their opinion based on that of their neighbours. If each agent adopts a positive opinion if and only if that opinion is sufficiently popular among his neighbours, the system will eventually settle into a fixed state or alternate between two states. If one agent acts in a different way, other periods may arise. We show that only a small number of periods may arise if natural restrictions are placed either on the neighbourhood structure or on the way in which the nonconforming agent may act; without either of these restrictions any period is possible.

math.CO↗

arXiv2015

Subdivisions in the Robber Locating Game

We consider a game in which a cop searches for a moving robber on a graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$ obtained by replacing each edge of $G$ by a path of length $m$, if $m \geqslant n$. They conjectured that this bound was best possible for complete graphs, but the present authors showed that in fact the cop wins on $K^{1/m}$ if and only if $m \geqslant n/2$, for all but a few small values of $n$. In this paper we extend this result to general graphs by proving that the cop has a winning strategy on $G^{1/m}$ provided $m \geqslant n/2$ for all but a few small values of $n$; this bound is best possible. We also consider replacing the edges of $G$ with paths of varying lengths.

math.CO↗

arXiv2015

Preferential attachment with choice

We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when $r=2$ and shows condensation-like behaviour when $r>2$.

math.PR↗

arXiv2014

Bounds on Herman's algorithm

Herman's self-stabilisation algorithm allows a ring of $N$ processors having any odd number of tokens to reach a stable state where exactly one token remains. McIver and Morgan conjecture that the expected time taken for stabilisation is maximised when there are three equally-spaced tokens. We prove exact results on a related cost function, and obtain a bound on expected time which is very close to the conjectured bound.

cs.DS↗

arXiv2014

The Robber Locating game

We consider a game in which a cop searches for a moving robber on a graph using distance probes, studied by Carragher, Choi, Delcourt, Erickson and West, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West show that for any fixed graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$, obtained by replacing each edge of $G$ by a path of length $m$, if $m$ is sufficiently large. They conjecture that the cop does not have a winning strategy on $K_n^{1/m}$ if $m<n$; we show that in fact the cop wins if and only if $m\geqslant n/2$, for all but a few small values of $n$. They also show that the robber can avoid capture on any graph of girth 3, 4 or 5, and ask whether there is any graph of girth 6 on which the cop wins. We show that there is, but that no such graph can be bipartite; in the process we give a counterexample for their conjecture that the set of graphs on which the cop wins is closed under the operation of subdividing edges. We also give a complete answer to the question of when the cop has a winning strategy on $K_{a,b}^{1/m}$.

math.CO↗

arXiv2013

Extremal results on average subtree density of series-reduced trees

Vince and Wang showed that the average subtree density of a series-reduced tree is between 1/2 and 3/4, answering a conjecture of Jamison. They ask under what conditions a sequence of such trees may have average subtree density tending to either bound; we answer these questions by giving simple necessary and sufficient conditions in each case.

math.CO↗

arXiv2009

Judicious partitions of 3-uniform hypergraphs

The vertices of any graph with $m$ edges can be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollobás and Thomason conjectured that the vertices of any $r$-uniform graph may be likewise partitioned into $r$ classes such that each part meets at least $cm$ edges, with $c=\frac{r}{2r-1}$. In this paper, we prove this conjecture for the case $r=3$. In the course of the proof we shall also prove an extension of the graph case which was conjectured by Bollobás and Scott.

math.CO↗