Correspondence Colouring and its Applications to List Colouring and Delay Colouring

In this thesis, we study correspondence colouring and its applications to list colouring and delay colouring. We give a detailed exposition of the paper of Dvořák, and Postle introducing correspondence colouring. Moreover, we generalize two important results in delay colouring. The first is a result by Georgakopoulos, stating that cubic graphs are 4-delay colourable. We show that delay colouring can be formulated as an instance of correspondence colouring. Then we show that the modified line graph of a cubic bipartite graph is generally 4-correspondence colourable, using a Brooks’ type theorem for correspondence colouring. This allows us to give a more simple proof of a stronger result. The second result is one by Edwards and Kennedy, which states that quartic bipartite graphs are 5-delay colourable. We introduce the notion of p-cyclic correspondence colouring which is a type of correspondence colouring that generalizes delay colouring. We then prove that the modified line graph of a quartic bipartite graph is 5-cyclic correspondence colourable using the Combinatorial Nullstellensatz. We also show that the maximum DP-chromatic number of any cycle plus triangles (CPT) graph is 4. We construct a CPT graph with DP-chromatic number at least 4. Moreover, the upper bound follows easily from the Brooks’ type theorem for correspondence colouring. Finally, we do a preliminary investigation into using parity techniques in correspondence colouring to prove that CPT graphs are 3-choosable

List colouring hypergraphs and extremal results for acyclic graphs

We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes. We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every $s$, every STS of high enough order is not $s$-list-colourable. The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability $p$ where the random graph $G_{n,p}$ asymptotically almost surely contains every tree of bounded degree. This $p$ is nearly optimal in terms of the maximum degree of the trees that we embed. Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph