Coloring problems in graph theory

In this thesis, we focus on variants of the coloring problem on graphs. A coloring of a graph $G$ is an assignment of colors to the vertices. A coloring is proper if no two adjacent vertices are assigned the same color. Colorings are a central part of graph theory and over time many variants of proper colorings have been introduced. The variants we study are packing colorings, improper colorings, and facial unique-maximum colorings. A packing coloring of a graph $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that the distance between any two vertices that receive color $i$ is greater than $i$. A $(d_1, \ldots, d_k)$-coloring of $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that the distance between any two vertices that receive color $i$ is greater than $d_i$. We study packing colorings of multi-layer hexagonal lattices, improving a result of Fiala, Klav\v{z}ar, and Lidick\\u27{y}, and find the packing chromatic number of the truncated square lattice. We also prove that subcubic planar graphs are $(1, 1, 2, 2, 2)$-colorable. A facial unique-maximum coloring of $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that no two adjacent vertices receive the same color and the maximum color on a face appears only once on that face. We disprove a conjecture of Fabrici and G\ {o}ring that plane graphs are facial unique-maximum $4$-colorable. Inspired by this result, we also provide sufficient conditions for the facial unique-maximum $4$-colorability of a plane graph. A $\{ 0, p \}$-coloring of $G$ is an assignment of colors $0$ and $p$ to the vertices of $G$ such that the vertices that receive color $0$ form an independent set and the vertices that receive color $p$ form a linear forest. We will explore $\{ 0, p \}$-colorings, an offshoot of improper colorings, and prove that subcubic planar $K_4$-free graphs are $\{ 0, p \}$-colorable. This result is a corollary of a theorem by Borodin, Kostochka, and Toft, a fact that we failed to realize before the completion of our proof

Acyclic edge-coloring using entropy compression

An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio

Coloring non-crossing strings

For a family of geometric objects in the plane $\mathcal{F}=\{S_1,\ldots,S_n\}$, define $\chi(\mathcal{F})$ as the least integer $\ell$ such that the elements of $\mathcal{F}$ can be colored with $\ell$ colors, in such a way that any two intersecting objects have distinct colors. When $\mathcal{F}$ is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most $k$ pseudo-disks, it can be proven that $\chi(\mathcal{F})\le 3k/2 + o(k)$ since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family $\mathcal{F}$ of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of $\mathcal{F}$ are only allowed to "touch" each other. Such a family is said to be $k$-touching if no point of the plane is contained in more than $k$ elements of $\mathcal{F}$. We give bounds on $\chi(\mathcal{F})$ as a function of $k$, and in particular we show that $k$-touching segments can be colored with $k+5$ colors. This partially answers a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems of strings.Comment: 19 pages. A preliminary version of this work appeared in the proceedings of EuroComb'09 under the title "Coloring a set of touching strings