Resumen de Coloring problems in cayley graphs
Javier Barajas Tomas
The general problem of finding the chromatic number of a given graph is as old as graph theory itself, An important part of the contemporary concepts or graph theory arose from the different attempts to solve the four-color problem, one of the most famous problems in the area. Problems related with graph colorings are still an active and prolific research area, see for instance the book of Jensen and Toft {\it Graph Coloring Problems} \cite{jt}, where the authors describe the state--of--the--art of more than two hundred open problems in the area. In particular, the problem of finding the chromatic number of the so--called distance graphs is mentioned in this book. This problem is one of the main motivations of the research developed in this thesis.
Distance graphs and circulant graphs are Cayley graphs (graphs with a transitive and regular group of automorphisms) of cyclic groups. The determination of the chromatic number of distance graphs and circulant graphs has attracted a considerable interest in the literature and it is one of the main topics of this Thesis. A frequently used method for coloring circulant graphs relates this problem with the so--called {\it Lonely Runner Problem\/} which is also one of the themes of this Thesis. One of the main contributions of this work is an algebraic tool that we call the Prime Filtering Lemma. It allows to distribute the elements through the ring of integers modulo $n$ by means of appropriated multipliers. This tool is intensively used to deal with chromatic problems in distance graphs and circulant graphs.
First we solve in the positive the first open case of a conjecture by Zhu (for degree 8) which states that a distance graph can have the maximum chromatic number only if it contains a large clique (in contrast with the situation for general graphs.) We also consider the next case (of degree 10), where the conjecture in its general form has been disproved, and we manage to give an almost complet c
