38 research outputs found
A note on vertex Tur\'an problems in the Kneser cube
The Kneser cube has vertex set and two vertices are
joined by an edge if and only if . For a fixed graph ,
we are interested in the most number of vertices of that span
a -free subgraph in . We show that the asymptotics of is
for bipartite and for graphs with chromatic
number at least 3. We also obtain results on the order of magnitude of
and in these two cases. In the case of
bipartite , we relate this problem to instances of the forbidden subposet
problem
Permutation codes
AbstractThere are many analogies between subsets and permutations of a set, and in particular between sets of subsets and sets of permutations. The theories share many features, but there are also big differences. This paper is a survey of old and new results about sets (and groups) of permutations, concentrating on the analogies and on the relations to coding theory. Several open problems are described
Improved covering results for conjugacy classes of symmetric groups via hypercontractivity
We study covering numbers of subsets of the symmetric group that
exhibit closure under conjugation, known as \emph{normal} sets. We show that
for any , there exists such that if and is a
normal subset of the symmetric group of density , then . This improves upon a seminal result of
Larsen and Shalev (Inventiones Math., 2008), with our in the double
exponent replacing their .
Our proof strategy combines two types of techniques. The first is
`traditional' techniques rooted in character bounds and asymptotics for the
Witten zeta function, drawing from the foundational works of Liebeck--Shalev,
Larsen--Shalev, and more recently, Larsen--Tiep. The second is a sharp
hypercontractivity theorem in the symmetric group, which was recently obtained
by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies
not only allows us to attain our improved bounds but also provides new insights
into the behavior of general independent sets in normal Cayley graphs over
symmetric groups
Sparse Kneser graphs are Hamiltonian
For integers and , the Kneser graph is the
graph whose vertices are the -element subsets of and whose
edges connect pairs of subsets that are disjoint. The Kneser graphs of the form
are also known as the odd graphs. We settle an old problem due to
Meredith, Lloyd, and Biggs from the 1970s, proving that for every ,
the odd graph has a Hamilton cycle. This and a known conditional
result due to Johnson imply that all Kneser graphs of the form
with and have a Hamilton cycle. We also prove that
has at least distinct Hamilton cycles for .
Our proofs are based on a reduction of the Hamiltonicity problem in the odd
graph to the problem of finding a spanning tree in a suitably defined
hypergraph on Dyck words
Research Problems from the BCC21
AbstractA collection of open problems, mostly presented at the problem session of the 21st British Combinatorial Conference
Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs
In this thesis we investigate three different aspects of graph theory.
Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs.
Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber.
Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses.
Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture