381 research outputs found
Covering and Separation for Permutations and Graphs
This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
A proof of the Ryser-Brualdi-Stein conjecture for large even
A Latin square of order is an by grid filled using symbols so
that each symbol appears exactly once in each row and column. A transversal in
a Latin square is a collection of cells which share no symbol, row or column.
The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every
Latin square of order contains a transversal with cells, and a
transversal with cells if is odd. Keevash, Pokrovskiy, Sudakov and
Yepremyan recently improved the long-standing best known bounds towards this
conjecture by showing that every Latin square of order has a transversal
with cells. Here, we show, for sufficiently large ,
that every Latin square of order has a transversal with cells.
We also apply our methods to show that, for sufficiently large , every
Steiner triple system of order has a matching containing at least
edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and
Yepremyan, who found such matchings with edges, and
proves a conjecture of Brouwer from 1981 for large .Comment: 71 pages, 13 figure
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Foundations of Node Representation Learning
Low-dimensional node representations, also called node embeddings, are a cornerstone in the modeling and analysis of complex networks. In recent years, advances in deep learning have spurred development of novel neural network-inspired methods for learning node representations which have largely surpassed classical \u27spectral\u27 embeddings in performance. Yet little work asks the central questions of this thesis: Why do these novel deep methods outperform their classical predecessors, and what are their limitations?
We pursue several paths to answering these questions. To further our understanding of deep embedding methods, we explore their relationship with spectral methods, which are better understood, and show that some popular deep methods are equivalent to spectral methods in a certain natural limit. We also introduce the problem of inverting node embeddings in order to probe what information they contain. Further, we propose a simple, non-deep method for node representation learning, and find it to often be competitive with modern deep graph networks in downstream performance.
To better understand the limitations of node embeddings, we prove some upper and lower bounds on their capabilities. Most notably, we prove that node embeddings are capable of exact low-dimensional representation of networks with bounded max degree or arboricity, and we further show that a simple algorithm can find such exact embeddings for real-world networks. By contrast, we also prove inherent bounds on random graph models, including those derived from node embeddings, to capture key structural properties of networks without simply memorizing a given graph
A general approach to transversal versions of Dirac-type theorems
Given a collection of hypergraphs with the same
vertex set, an -edge graph is a transversal if
there is a bijection such that for
each . How large does the minimum degree of each need to be so
that necessarily contains a copy of that is a transversal?
Each in the collection could be the same hypergraph, hence the minimum
degree of each needs to be large enough to ensure that .
Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020,
52(3):498-504], a growing body of work has shown that in many cases this lower
bound is tight. In this paper, we give a unified approach to this problem by
providing a widely applicable sufficient condition for this lower bound to be
asymptotically tight. This is general enough to recover many previous results
in the area and obtain novel transversal variants of several classical
Dirac-type results for (powers of) Hamilton cycles. For example, we derive that
any collection of graphs on an -vertex set, each with minimum degree at
least , contains a transversal copy of the -th power of a
Hamilton cycle. This can be viewed as a rainbow version of the P\'osa-Seymour
conjecture.Comment: 21 pages, 4 figures; final version as accepted for publication in the
Bulletin of the London Mathematical Societ
Counting spanning subgraphs in dense hypergraphs
We give a simple method to estimate the number of distinct copies of some
classes of spanning subgraphs in hypergraphs with high minimum degree. In
particular, for each and , we show that every
-graph on vertices with minimum codegree at least
\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\
\left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if
$(k-\ell)\nmid k$,}
contains Hamilton -cycles as long as
. When this gives a simple proof of a result
of Glock, Gould, Joos, K\"uhn and Osthus, while, when this
gives a weaker count than that given by Ferber, Hardiman and Mond or, when
, by Ferber, Krivelevich and Sudakov, but one that holds for an
asymptotically optimal minimum codegree bound
Influence Maximization based on Simplicial Contagion Model in Hypergraph
In recent years, the issue of node centrality has been actively and
extensively explored due to its applications in product recommendations,
opinion propagation, disease spread, and other areas involving maximizing node
influence. This paper focuses on the problem of influence maximization on the
Simplicial Contagion Model, using the susceptible-infectedrecovered (SIR) model
as an example. To find practical solutions to this optimization problem, we
have developed a theoretical framework based on message passing processes and
conducted stability analysis of equilibrium solutions for the self-consistent
equations. Furthermore, we introduce a metric called collective influence and
propose an adaptive algorithm, known as the Collective Influence Adaptive
(CIA), to identify influential propagators in the spreading process. This
method has been validated on both synthetic hypergraphs and real hypergraphs,
outperforming other competing heuristic methods.Comment: 19 pages,16 figure
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Paths and cycles in graphs and hypergraphs
In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles.
A -uniform tight cycle is a -uniform hypergraph on vertices with a cyclic ordering of its vertices such that the edges are all -sets of consecutive vertices in the ordering.
We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to -uniform hypergraphs and prove results in the 4- and 5-uniform case.
For a -uniform hypergraph~, the Ramsey number is the smallest integer such that any 2-edge-colouring of the complete -uniform hypergraph on vertices contains a monochromatic copy of . We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that = (5+(1)).
We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any >0 and 3 asymptotically almost surely, every subgraph of the binomial random -uniform hypergraph in which all -sets are contained in at least edges has a tight Hamilton cycle.
A random graph model on a host graph is said to be 1-independent if for every pair of vertex-disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . We show that = 4 - 2 is the critical probability such that every 1-independent graph model on where each edge is present with probability at least contains an infinite path
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