CLOSURE UNDER THE MAJORIZATION RELATION AND THE DISTINGUISHING CHROMATIC NUMBER OF CIRCULANT GRAPHS

Abstract

This dissertation addresses two distinct problems in graph theory and in each case advances results for invariants of graphs. The first problem investigates the arrangement of the degree sequences of various classes of graphs in the dominance order. In the second problem we consider a family of graphs, best described as a generalization of cycle graphs, and investigate the values of the distinguishing chromatic number for the complements of these graphs. An improvement of a known bound for this number is also given for this particular class of graphs. Chapter 1 introduces fundamental definitions, concepts, notations and known results that are used throughout the rest of the thesis. In Chapter 2, we take a look at how degree sequences of various classes of graphs are ordered by the majorization relation. This ordering gives rise to a poset known as the dominance order within which the degree sequences of threshold and split graphs were shown by Hammer et al. and Merris to display an orderly arrangement. To give context to these examples and better understand how degree sequences of classes of graphs situate themselves in the dominance order, we define what it means for a collection of graphs F to be dominance monotone. Furthermore, we characterize the dominance monotone sets F for “small" families F, and as a result two new classes of graphs whose degree sequences form an upward-closed set in the dominance order are discovered and identified. Chapter 3 is dedicated to the study of the distinguishing chromatic number for the complements of circulant graphs Cn(1; k), a family of graphs formed by adding a set of chords to a cycle. In general, we use our knowledge of the graph structure, and known and proven results such as the automorphism group of the graph to come up with constructions that determine the distinguishing chromatic number of their complements. These results, together with previous results on the distinguishing chromatic number for the circulant graphs Cn(1; k) from a joint work done with Barrus and Lantz, provide an improvement for upper bounds on the sum and product of the distinguishing chromatic numbers of these graphs and their complements produced by Collins and Trenk.