18 research outputs found
A collection of open problems in celebration of Imre Leader's 60th birthday
One of the great pleasures of working with Imre Leader is to experience his
infectious delight on encountering a compelling combinatorial problem. This
collection of open problems in combinatorics has been put together by a subset
of his former PhD students and students-of-students for the occasion of his
60th birthday. All of the contributors have been influenced (directly or
indirectly) by Imre: his personality, enthusiasm and his approach to
mathematics. The problems included cover many of the areas of combinatorial
mathematics that Imre is most associated with: including extremal problems on
graphs, set systems and permutations, and Ramsey theory. This is a personal
selection of problems which we find intriguing and deserving of being better
known. It is not intended to be systematic, or to consist of the most
significant or difficult questions in any area. Rather, our main aim is to
celebrate Imre and his mathematics and to hope that these problems will make
him smile. We also hope this collection will be a useful resource for
researchers in combinatorics and will stimulate some enjoyable collaborations
and beautiful mathematics
Extremal Problems on the Hypercube
PhDThe hypercube, Qd, is a natural and much studied combinatorial object, and we discuss
various extremal problems related to it.
A subgraph of the hypercube is said to be (Qd; F)-saturated if it contains no copies of
F, but adding any edge forms a copy of F. We write sat(Qd; F) for the saturation number,
that is, the least number of edges a (Qd; F)-saturated graph may have. We prove the
upper bound sat(Qd;Q2) < 10 2d, which strongly disproves a conjecture of Santolupo that
sat(Qd;Q2) =
�� 1 4 + o(1)
d2d��1. We also prove upper bounds on sat(Qd;Qm) for general
m.Given a down-set A and an up-set B in the hypercube, Bollobás and Leader conjectured
a lower bound on the number of edge-disjoint paths between A and B in the directed
hypercube. Using an unusual form of the compression argument, we confirm the conjecture
by reducing the problem to a the case of the undirected hypercube. We also prove an
analogous conjecture for vertex-disjoint paths using the same techniques, and extend both
results to the grid.
Additionally, we deal with subcube intersection graphs, answering a question of Johnson
and Markström of the least r = r(n) for which all graphs on n vertices may be represented as
subcube intersection graph where each subcube has dimension exactly r. We also contribute
to the related area of biclique covers and partitions, and study relationships between various
parameters linked to such covers and partitions.
Finally, we study topological properties of uniformly random simplicial complexes, employing
a characterisation due to Korshunov of almost all down-sets in the hypercube as a
key tool
Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)
International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..
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
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum