Searcharxiv⌕ Search

arXiv subjects

Alex Scott

Publications and source records attributed to Alex Scott.

162 records · Page 9Linked to original sources
arXiv2016

On a problem of Erdos and Moser

A set $A$ of vertices in an $r$-uniform hypergraph $\mathcal H$ is covered in $\mathcal H$ if there is some vertex $u\not\in A$ such that, for every $(r-1)$-set $B\subset A$, the set $\{u\}\cup B$ is in $\mathcal H$. Erdos and Moser (1970) determined the minimum number of edges in a graph on $n$ vertices such that every $k$-set is covered. We extend this result to $r$-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.

math.CO↗
arXiv2016

Packing random graphs and hypergraphs

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have different densities. In the graph case, we prove a stronger result, on packing a random graph with a fixed graph.

math.CO↗
arXiv2016

Feedback from Nature: Simple Randomised Distributed Algorithms for Maximal Independent Set Selection and Greedy Colouring

We propose distributed algorithms for two well-established problems that operate efficiently under extremely harsh conditions. Our algorithms achieve state-of-the-art performance in a simple and novel way. Our algorithm for maximal independent set selection operates on a network of identical anonymous processors. The processor at each node has no prior information about the network. At each time step, each node can only broadcast a single bit to all its neighbours, or remain silent. Each node can detect whether one or more neighbours have broadcast, but cannot tell how many of its neighbours have broadcast, or which ones. We build on recent work of Afek et al. which was inspired by studying the development of a network of cells in the fruit fly~\cite{Afek2011a}. However we incorporate for the first time another important feature of the biological system: varying the probability value used at each node based on local feedback from neighbouring nodes. Given any $n$-node network, our algorithm achieves the optimal expected time complexity of $O(\log n)$ rounds and the optimal expected message complexity of $O(1)$ single-bit messages broadcast by each node.We also show that the previous approach, without feedback, cannot achieve better than $Ω(\log^2 n)$ expected time complexity, whatever global scheme is used to choose the probabilities. Our algorithm for distributed greedy colouring works under similar harsh conditions: each identical node has no prior information about the network, can only broadcast a single message to all neighbours at each time step representing a desired colour, and can only detect whether at least one neighbour has broadcast each colour value. We show that our algorithm has an expected time complexity of $O(Δ+\log n)$, where $Δ$ is the maximum degree of the network, and expected message complexity of $O(1)$ messages broadcast by each node.

cs.DC↗
arXiv2015

Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyarfas' conjectures

Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chromatic number contains either a 5-hole or a hole of length more than $l$.

math.CO↗
arXiv2015

Saturation in the Hypercube and Bootstrap Percolation

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $Θ(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.

math.CO↗
arXiv2014

Disjoint dijoins

A dijoin in a digraph is a set of edges meeting every directed cut. D. R. Woodall conjectured in 1976 that if G is a digraph, and every directed cut of G has at least k edges, then there are k pairwise disjoint dijoins. This remains open, but a capacitated version is known to be false. In particular, A. Schrijver gave a digraph G and a subset S of its edge-set, such that every directed cut contains at least two edges in S, and yet there do not exist two disjoint dijoins included in S. In Schrijver's example, G is planar, and the subdigraph formed by the edges in S consists of three disjoint paths. We conjecture that when k = 2, the disconnectedness of S is crucial: more precisely, that if G is a digraph, and S is a subset of the edges of G that forms a connected subdigraph (as an undirected graph), and every directed cut of G contains at least two edges in S, then we can partition S into two dijoins. We prove this in two special cases: when G is planar, and when the subdigraph formed by the edges in S is a subdivision of a caterpillar.

math.CO↗
arXiv2014

Disjoint paths in tournaments

Given $k$ pairs of vertices $(s_i,t_i)$, $1\le i\le k$, of a digraph $G$, how can we test whether there exist $k$ vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but for $k=2$ there is a polynomial-time algorithm when $G$ is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen. Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $G$ is semicomplete.

math.CO↗
arXiv2014

Intersections of hypergraphs

Given two weighted k-uniform hypergraphs G, H of order n, how much (or little) can we make them overlap by placing them on the same vertex set? If we place them at random, how concentrated is the distribution of the intersection? The aim of this paper is to investigate these questions.

math.CO↗
arXiv2014

On Saturated $k$-Sperner Systems

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$. A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.

math.CO↗
arXiv2013

Hypergraphs of bounded disjointness

A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner, Lemons, Palmer, Patkós and Szécsi conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost intersecting hypergraph has at most $(s+1)\binom{2k-2}{k-1}$ edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related results and conjectures.

math.CO↗
arXiv2013

Substitution and $χ$-Boundedness

A class $\mathcal{G}$ of graphs is said to be {\em $χ$-bounded} if there is a function $f:\mathbb{N} \rightarrow \mathbb{R}$ such that for all $G \in \mathcal{G}$ and all induced subgraphs $H$ of $G$, $χ(H) \leq f(ω(H))$. In this paper, we show that if $\mathcal{G}$ is a $χ$-bounded class, then so is the closure of $\mathcal{G}$ under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if $\mathcal{G}$ is $χ$-bounded by a polynomial (respectively: exponential) function, then the closure of $\mathcal{G}$ under substitution is also $χ$-bounded by some polynomial (respectively: exponential) function. In addition, we show that if $\mathcal{G}$ is a $χ$-bounded class, then the closure of $\mathcal{G}$ under the operations of gluing along a clique and gluing along a bounded number of vertices together is also $χ$-bounded, as is the closure of $\mathcal{G}$ under the operations of substitution and gluing along a clique together.

math.CO↗
arXiv2013

Excluding Pairs of Graphs

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$, we say that $G$ {\em admits an $(\mathcal{F},P)$-partition} if the vertex set of $G$ can be partitioned into $P$ subsets $X_1,..., X_P$, so that for every $i \in \{1,..., P\}$, either $|X_i|=1$, or the subgraph of $G$ induced by $X_i$ is $\{F\}$-free for some $F \in \mathcal{F}$. Our first result is the following. For every pair $(H,J)$ of graphs such that $H$ is the disjoint union of two graphs $H_1$ and $H_2$, and the complement $J^c$ of $J$ is the disjoint union of two graphs $J_1^c$ and $J_2^c$, there exists an integer $P>0$ such that every $\{H,J\}$-free graph has an $(\{H_1,H_2,J_1,J_2\},P)$-partition. Using a similar idea we also give a short proof of one of the results of \cite{heroes}. Our final result is a construction showing that if $\{H,J\}$ are graphs each with at least one edge, then for every pair of integers $r,k$ there exists a graph $G$ such that every $r$-vertex induced subgraph of $G$ is $\{H,J\}$-split, but $G$ does not admits an $(\{H,J\},k)$-partition.

math.CO↗
arXiv2012

Feedback from nature: an optimal distributed algorithm for maximal independent set selection

Maximal Independent Set selection is a fundamental problem in distributed computing. A novel probabilistic algorithm for this problem has recently been proposed by Afek et al, inspired by the study of the way that developing cells in the fly become specialised. The algorithm they propose is simple and robust, but not as efficient as previous approaches: the expected time complexity is O(log^2 n). Here we first show that the approach of Afek et al cannot achieve better efficiency than this across all networks, no matter how the probability values are chosen. However, we then propose a new algorithm that incorporates another important feature of the biological system: adapting the probabilities used at each node based on local feedback from neighbouring nodes. Our new algorithm retains all the advantages of simplicity and robustness, but also achieves the optimal efficiency of O(log n) expected time.

cs.DC↗
arXiv2012

Cover-Decomposition and Polychromatic Numbers

A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VC-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover.

math.CO↗
arXiv2011

Monochromatic cycles and the monochromatic circumference in 2-coloured graphs

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with minimal degree equal to 3n/4 that do not contain all such cycles. Finally we show that, for all positive constants d and sufficiently large n, a 2-edge coloured graph G of order n with minimal degree at least 3n/4 either contains a monochromatic cycle of length at least (2/3+d/2)n, or, in one of the two colours, contains a cycle of all lengths between three and (2/3-d)n.

math.CO↗
arXiv2010

A new bound for the cops and robbers problem

In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph $G$. Cops and a robber are allowed to move along the edges of $G$ and the goal of cops is to capture the robber. The cop number $c(G)$ of $G$ is the minimum number of cops required to win the game. Meyniel conjectured a long time ago that $O(\sqrt{n})$ cops are enough for any connected $G$ on $n$ vertices. Improving several previous results, we prove that the cop number of $n$-vertex graph is at most $n 2^{-(1+o(1))\sqrt{\log n}}$.

math.CO↗