On choosability and paintability of graphs

Let $G=(V,E)$ be a graph. A \emph{list assignment} $L$ for $G$ is a function from

$V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$

with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$

whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list

assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$

there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called

the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite

graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al.

suggested the problem of determining $\ensuremath{\ch(K_{s*k})}$, and showed that

$\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved

the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that

$\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$.

An online version of the list coloring problem was introduced independently by Schauz

and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice

designs lists of colors for all vertices, but does not tell Bob, and is allowed to

change her mind about unrevealed colors as the game progresses. On her $i$-th turn

Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides,

irrevocably, which (independent set) of these vertices to color with $i$. For a

function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if

eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its

list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online

choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If

$l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable.

The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$

is $t$-paintable, and is denoted by $\ensuremath{\ch^{\mathrm{OL}}(G)}$. Evidently,

$\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$

can be arbitrarily large. However there are only a few known examples with this gap

equal to one, and none with larger gap. This conjecture is explored in this thesis.

One of the obstacles is that there are not many graphs whose exact list coloring

number is known. This is one of the motivations for establishing new cases of Erd\H{o}s'

problem. Here new examples of graphs with gap one are found, and related technical

results are developed as tools for attacking Zhu's conjecture.

The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices

at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$.

This was recently disproved for specially constructed graphs. Here it is shown that

a graph arising naturally in the theory of cellular networks is also a counterexample.