Bounds on the Defective, Multifold, Paint Number of Planar Graphs

A $k$-list assignment for a graph $G=(V, E)$ is a function $L$ that assigns a $k$-set $L(v)$ of "available colors" to each vertex $v \in V$. A $d$-defective, $m$-fold, $L$-coloring is a function $\phi$ that assigns an $m$-subset $\phi(v) \subseteq L(v)$ to each vertex $v$ so that each color class $V_{i}=\{v \in V:$ $i \in \phi(v)\}$ induces a subgraph of $G$ with maximum degree at most $d$. An edge $xy$ is an $i$-flaw of $\phi$ if $i\in \phi(x) \cap \phi(y)$. An online list-coloring algorithm $\mathcal{A}$ works on a known graph $G$ and an unknown $k$-list assignment $L$ to produce a coloring $\phi$ as follows. At step $r$ the set of vertices $v$ with $r \in L(v)$ is revealed to $\mathcal{A}$. For each vertex $v$, $\mathcal{A}$ must decide irrevocably whether to add $r$ to $\phi(v)$. The online choice number $\pt_{m}^{d}(G)$ of $G$ is the least $k$ for which some such algorithm produces a $d$-defective, $m$-fold, $L$-coloring $\phi$ of $G$ for all $k$-list assignments $L$. Online list coloring was introduced independently by Uwe Schauz and Xuding Zhu. It was known that if $G$ is planar then $\pt_{1}^{0}(G) \leq 5$ and $\pt_{1}^{1}(G) \leq 4$ are sharp bounds; here it is proved that $\pt_{1}^{3}(G) \leq 3$ is sharp, but there is a planar graph $H$ with $\pt_{1}^{2}(H)\ge 4$. Zhu conjectured that for some integer $m$, every planar graph $G$ satisfies $\pt_{m}^{0}(G) \leq 5 m-1$, and even that this is true for $m=2$. This dissertation proves that $\pt_{2}^{1}(G) \leq 9$, so the conjecture is "nearly" true, and the proof extends to $\pt_{m}^{1}(G) \leq\left\lceil\frac{9}{2} m\right\rceil$. Using Alon's Combinatorial Nullstellensatz, this is strengthened by showing that $G$ contains a linear forest $(V, F)$ such that there is an online algorithm that witnesses $\mathrm{pt}_{2}^{1}(G) \leq 9$ while producing a coloring whose flaws are in $F$, and such that no edge is an $i$-flaw and a $j$-flaw for distinct colors $i$ and $j$.