Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

Cycle and Cocycle Coverings of Graphs

Abstract: In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family C of at most n−1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and edges having cogirth g * ≥ 3 and k(G) components, there is a family of at most −n+k(G) cocycles which cover the edges of G at least twice

Aspects of graph colouring

The four-colour conjecture of 1852, and the total colouring conjecture of 1965, have sparked off many new concepts and conjectures. In this thesis we investigate many of the outstanding conjectures, establishing various related results, and present many conjectures of our own. We give a brief historical introduction (Chapter 1) and establish some notation, terminology and techniques (Chapter 2). Next, in Chapter 3, we examine the use of latin squares to represent edge and total colourings. In Chapters 4 - 6 we deal with vertex, edge and total colourings respectively. Various ways of measuring different aspects of graphs are presented, in particular, the ‘colouring difference’ between two edge-colourings of a graph (Chapter 5) and the ‘beta parameter’ (defined in Chapter 2 and used in Chapters 3 and 6); this is a measure of how far from a type 1 graph a type 2 graph can be. In Chapter 6 we derive an upper bound for the beta value of any near type 1 graph and give the exact results for all Kn. The number of ways of colouring Kn and Kn,,n are also quantified. Chapter 6 also examines Hilton’s concept of conformability. It is shown that every graph with at least A spines is conformable, and an extension to the concept, which we call G*-conformability, is introduced. We then give new necessary conditions for a cubic graph to be type 1 in relation to G*-conformability. Various methods of manipulating graphs are considered and we present: a method to compatibly triangulate a graph G-e; a method of introducing a fourth colour thus allowing a sequence of Kempe interchanges from any edge 3-colouring of a cubic graph to any other; and a method to re-colour a near type 1 graph within a certain bound on beta. We end this thesis with a brief discussion on possible practical uses for colouring graphs. A list of the main results and conjectures is given at the end of each chapter, but a short list of the principle theorems proven is given below

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

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

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum