The Local Chromatic Number

A graph vertex colouring is called k-local if the number of colours used in the closed neighbourhood of each vertex is at most k. The local chromatic number of a graph is the smallest k for which the graph has a proper k-local colouring. So unlike the chromatic number which is the minimum total number of colours required in a proper colouring, the local chromatic number is minimum number of colours that must appear in the closed neighbourhood of some vertex in a proper colouring. In this thesis we will examine basic properties of the local chromatic number, and techniques used to determine or bound it. We will examine a theory that was sparked by Lovász's original proof of the Kneser conjecture, using topological tools to give lower bounds on the chromatic number, and see how it is applicable to give lower bounds on the local chromatic number as well. The local chromatic number lies between the fractional chromatic number and the chromatic number, and thus it is particularly interesting to study when the gap between these two parameters is large. We will examine the local chromatic number for specific classes of graphs, and give a slight generalization of a result by Simonyi and Tardos that gives an upper bound on the local chromatic number for a class of graphs called Schrijver graphs. Finally we will discuss open conjectures about the chromatic number and investigate versions adapted to the local chromatic number

Optimal k-fold colorings of webs and antiwebs

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by \chi_k(G). We determine the exact value of this parameter when G is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which \chi_k(G) attains its lower and upper bounds based on the clique, the fractional chromatic and the chromatic numbers. Additionally, we extend the concept of \chi-critical graphs to \chi_k-critical graphs. We identify the webs and antiwebs having this property, for every integer k <= 1.Comment: A short version of this paper was presented at the Simp\'osio Brasileiro de Pesquisa Operacional, Brazil, 201