Search results

(1 - 1 of 1)
ON THE LIST COLORING PROBLEM AND ITS EQUITABLE VARIANTS
Title
ON THE LIST COLORING PROBLEM AND ITS EQUITABLE VARIANTS
Creator
Mudrock, Jeffrey Allen
Date
2018, 2018-05
Description
In this thesis we study list coloring which was introduced independently by Vizing and Erd˝os, Rubin, and Taylor in the 1970’s. Suppose we...
Show moreIn this thesis we study list coloring which was introduced independently by Vizing and Erd˝os, Rubin, and Taylor in the 1970’s. Suppose we associate a list assignment L with a graph G which assigns a list, L(v), of colors to each v 2 V (G). A proper L-coloring of G, f, is a proper coloring such that f(v) 2 L(v) for each v 2 V (G). The list chromatic number of G, "`(G), is the minimum k such that G has a proper L-coloring whenever L is a list assignment satisfying |L(v)| ' k for each v 2 V (G). A graph G is said to be chromatic-choosable if "`(G) = "(G). The list chromatic number of the Cartesian product of graphs is not well understood. The best result is by Borowiecki, Jendrol, Kr´al, and Miˇskuf (2006) who proved that the list chromatic number of the Cartesian product of two graphs can be bounded in terms of the list chromatic number and the coloring number of the factors. In Chapter 2, we use the Alon-Tarsi Theorem and an extension of it discovered by Schauz in 2010 to find improved bounds on the list chromatic number and paint number (i.e. online list chromatic number) of the Cartesian product of an odd cycle or complete graph with a traceable graph. We also identify certain Cartesian products as chromatic-choosable. In Chapter 3, we generalize the notion of strong critical graphs, introduced by Stiebitz, Tuza, and Voigt in 2008, to strong k-chromatic-choosable graphs, and we show that it gives a strictly larger family of graphs that includes odd cycles, cliques, the join of a clique and any strongly chromatic-choosable graph, and many more families of graphs. We prove sharp bounds on the list chromatic number of certain Cartesian products where one factor is a strong k-chromatic-choosable graph satisfying an edge bound. Our proofs rely on the notion of unique-choosability as a sufficient condition for list colorability and the list color function which is a list analogue of the chromatic polynomial.In Chapter 4, we study a list analogue of equitable coloring introduced by Kostochka, Pelsmajer, and West in 2003. A graph G is said to be equitably kchoosable if it has a proper L-coloring that uses no color more than d|V (G)|/ke times whenever |L(v)| = k for each v 2 V (G). Generalizing a conjecture of Fu (1994) on total equitable coloring, we conjecture that for any simple graph G, its total graph, T(G), is equitably k-choosable whenever k ' max{"`(T(G)),"(G) + 2}. We prove this conjecture for all graphs satisfying "(G)  2 while also studying the related question of the equitable choosability of powers of paths and cycles. In Chapter 5, we introduce a new list analogue of equitable coloring: proportional choosability. For this new notion, the number of times a color is used must be proportional to the number of lists in which the color appears. Proportional kchoosability implies both equitable k-choosability and equitable k-colorability. Also,the graph property of being proportionally k-choosable is monotone, and if a graph is proportionally k-choosable, it must be proportionally (k +1)-choosable. We study the proportional choosability of graphs with small order and disconnected graphs, and we completely characterize proportionally 2-choosable graphs.
Ph.D. in Applied Mathematics, May 2018
Show less