On List-Coloring and the Sum List Chromatic Number of Graphs.

This thesis explores several of the major results in list-coloring in an expository fashion. As a specialization of list coloring, the sum list chromatic number is explored in detail. Ultimately, the thesis is designed to motivate the discussion of coloring problems and, hopefully, interest the reader in the branch of coloring problems in graph theory

An upper bound on the fractional chromatic number of triangle-free subcubic graphs

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The fractional chromatic number $\chi_f(G)$ is the infimum of $a/b$ over all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami and Zhu proved that $\chi_f(G) \leq 3-3/64 \approx 2.953$. Lu and Peng improved the bound to $\chi_f(G) \leq 3-3/43 \approx 2.930$. Recently, Ferguson, Kaiser and Kr\'{a}l' proved that $\chi_f(G) \leq 32/11 \approx 2.909$. In this paper, we prove that $\chi_f(G) \leq 43/15 \approx 2.867$

Extensions of Fractional Precolorings show Discontinuous Behavior

We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\ge3 and any d. We determine the exact values of epsilon for k \in (2,3) if d=4, and k \in [2.5,3) if d=6, and give upper bounds for k \in (2,3) if d=5,7, and k \in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d

A Branch and Price Algorithm for List Coloring Problem

Coloring problems in graphs have been used to model a wide range of real applications. In particular, the List Coloring Problem generalizes the well-known Graph Coloring Problem for which many exact algorithms have been developed. In this work, we present a Branch-and-Price algorithm for the weighted version of the List Coloring Problem, based on the one developed by Mehrotra and Trick (1996) for the Graph Coloring Problem. This version considers non-negative weights associated to each color and it is required to assign a color to each vertex from predetermined lists in such a way the sum of weights of the assigned colors is minimum. Computational experiments show the good performance of our approach, being able to comfortably solve instances whose graphs have up to seventy vertices. These experiences also bring out that the hardness of the instances of the List Coloring Problem does not seem to depend only on quantitative parameters such as the size of the graph, its density, and the size of list of colors, but also on the distribution of colors present in the lists.Fil: Lucci, Mauro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Severin, Daniel Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina10th Latin and American Algorithms, Graphs and Optimization Symposium (LAGOS 2019)Belo HorizonteBrasilCoordenação de Aperfeiçoamento de Pessoal de Nivel SuperiorConselho Nacional de Desenvolvimento Científico e Técnologico do BrasilUniversidade Federal de Minas Gerai