Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Colouring random graphs: Tame colourings

Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded chromatic number of G, denoted by $\chi_t(G)$, is the minimum number of colours in such a colouring. Every colouring of G is then $\alpha(G)$-bounded, where $\alpha(G)$ denotes the size of a largest independent set. We study colourings of the random graph G(n, 1/2) and of the corresponding uniform random graph G(n,m) with $m=\left \lfloor \frac 12 {n \choose 2} \right \rfloor$. We show that $\chi_t(G(n,m))$ is maximally concentrated on at most two explicit values for $t = \alpha(G(n,m))-2$. This behaviour stands in stark contrast to that of the normal chromatic number, which was recently shown not to be concentrated on any sequence of intervals of length $n^{1/2-o(1)}$. Moreover, when $t = \alpha(G_{n, 1/2})-1$ and if the expected number of independent sets of size $t$ is not too small, we determine an explicit interval of length $n^{0.99}$ that contains $\chi_t(G_{n,1/2})$ with high probability. Both results have profound consequences: the former is at the core of the intriguing Zigzag Conjecture on the distribution of $\chi(G_{n, 1/2})$ and justifies one of its main hypotheses, while the latter is an important ingredient in the proof of a non-concentration result for $\chi(G_{n,1/2})$ that is conjectured to be optimal. These two results are consequences of a more general statement. We consider a class of colourings that we call tame, and provide tight bounds for the probability of existence of such colourings via a delicate second moment argument. We then apply those bounds to the two aforementioned cases. As a further consequence of our main result, we prove two-point concentration of the equitable chromatic number of G(n,m).Comment: 75 page

Configurations and colouring problems in block designs

A Steiner triple system of order v (STS(v)) is called x-chromatic if x is the smallest number of colours needed to avoid monochromatic blocks. Amongst our results on colour class structures we show that every STS (19) is 3- or 4-chromatic, that every 3-chromatic STS(19) has an equitable 3-colouring (meaning that the colours are as uniformly distributed as possible), and that for all admissible v > 25 there exists a 3-chromatic STS(v) which does not admit an equitable 3-colouring. We obtain a formula for the number of independent sets in an STS(v) and use it to show that an STS(21) must contain eight independent points. This leads to a simple proof that every STS(21) is 3- or 4-chromatic. Substantially extending existing tabulations, we provide an enumeration of STS trades of up to 12 blocks, and as an application we show that any pair of STS(15)s must be 3-1-isomorphic. We prove a general theorem that enables us to obtain formulae for the frequencies of occurrence of configurations in triple systems. Some of these are used in our proof that for v > 25 no STS(u) has a 3-existentially closed block intersection graph. Of specific interest in connection with a conjecture of Erdos are 6-sparse and perfect Steiner triple systems, characterized by the avoidance of specific configurations. We describe two direct constructions that produce 6-sparse STS(v)s and we give a recursive construction that preserves 6-sparseness. Also we settle an old question concerning the occurrence of perfect block transitive Steiner triple systems. Finally, we consider Steiner 5(2,4, v) designs that are built from collections of Steiner triple systems. We solve a longstanding problem by constructing such systems with v = 61 (Zoe’s design) and v = 100 (the design of the century)

Equitably coloured balanced incomplete block designs

In this thesis we determine necessary and sufficient conditions for the existence of an equitably ℓ-colourable balanced incomplete block design for any positive integer ℓ > 2. In particular, we present a method for constructing non-trivial equitably ℓ-colourable BIBDs and prove that these designs are the only non-trivial equitably ℓ-colourable BIBDs that exist. We also observe that every equitable ℓ-colouring of a BIBD yields both an equalised ℓ-colouring and a proper 2-colouring of the same BIBD. We also discuss generalisations of these concepts including open questions for further research. The main results presented in this thesis also appear in [7]