research
Online choosability of graphs
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Abstract
We study several problems in graph coloring. In list coloring, each vertex v has a set L(v) of available colors and must be assigned a color from this set so that adjacent vertices receive distinct colors; such a coloring is an L-coloring, and we then say that G is L-colorable. Given a graph G and a function f:V(G)→N, we say that G is f-choosable if G is L-colorable for any list assignment L such that ∣L(v)∣≥f(v) for all v∈V(G). When f(v)=k for all v and G is f-choosable, we say that G is k-choosable. The least k such that G is k-choosable is the choice number, denoted ch(G). We focus on an online version of this problem, which is modeled by the Lister/Painter game.
The game is played on a graph in which every vertex has a positive number of tokens. In each round, Lister marks a nonempty subset M of uncolored vertices, removing one token at each marked vertex. Painter responds by selecting a subset D of M that forms an independent set in G. A color distinct from those used on previous rounds is given to all vertices in D. Lister wins by marking a vertex that has no tokens, and Painter wins by coloring all vertices in G. When Painter has a winning strategy, we say that G is f-paintable. If f(v)=k for all v and G is f-paintable, then we say that G is k-paintable. The least k such that G is k-paintable is the paint number, denoted \pa(G).
In Chapter 2, we develop useful tools for studying the Lister/Painter game. We study the paintability of graph joins and of complete bipartite graphs. In particular, \pa(K_{k,r})\le k if and only if r<kk.
In Chapter 3, we study the Lister/Painter game with the added restriction that the proper coloring produced by Painter must also satisfy some property P. The main result of Chapter 3 provides a general method to give a winning strategy for Painter when a strategy for the list coloring problem is already known. One example of a property P is that of having an r-dynamic coloring, where a proper coloring is r-dynamic if each vertex v has at least min{r,d(v)} distinct colors in its neighborhood. For any graph G and any r, we give upper bounds on how many tokens are necessary for Painter to produce an r-dynamic coloring of G. The upper bounds are in terms of r and the genus of a surface on which G embeds.
In Chapter 4, we study a version of the Lister/Painter game in which Painter must assign m colors to each vertex so that adjacent vertices receive disjoint color sets. We characterize the graphs in which 2m tokens is sufficient to produce such a coloring. We strengthen Brooks' Theorem as well as Thomassen's result that planar graphs are 5-choosable.
In Chapter 5, we study sum-paintability. The sum-paint number of a graph G, denoted \spa(G), is the least ∑f(v) over all f such that G is f-paintable. We prove the easy upper bound: \spa(G)\le|V(G)|+|E(G)|. When \spa(G)=|V(G)|+|E(G)|, we say that G is sp-greedy. We determine the sum-paintability of generalized theta-graphs. The generalized theta-graph Θℓ1,…,ℓk consists of two vertices joined by k paths of lengths \VEC \ell1k. We conjecture that outerplanar graphs are sp-greedy and prove several partial results toward this conjecture.
In Chapter 6, we study what happens when Painter is allowed to allocate tokens as Lister marks vertices. The slow-coloring game is played by Lister and Painter on a graph G. Lister marks a nonempty set of uncolored vertices and scores 1 point for each marked vertex. Painter colors all vertices in an independent subset of the marked vertices with a color distinct from those used previously in the game. The game ends when all vertices have been colored. The sum-color cost of a graph G, denoted \scc(G), is the maximum score Lister can guarantee in the slow-coloring game on G. We prove several general lower and upper bounds for \scc(G). In more detail, we study trees and prove sharp upper and lower bounds over all trees with n vertices. We give a formula to determine \scc(G) exactly when α(G)≤2. Separately, we prove that \scc(G)=\spa(G) if and only if G is a disjoint union of cliques. Lastly, we give lower and upper bounds on \scc(K_{r,s})