Beyond Ohba's Conjecture: A bound on the choice number of kk-chromatic graphs with nn vertices

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend this to a general upper bound: $\text{ch}(G)\le \max\{\chi(G),\lceil({|V(G)|+\chi(G)-1})/{3}\rceil\}$. Our result is sharp for $|V(G)|\le 3\chi(G)$ using Ohba's examples, and it improves the best-known upper bound for $\text{ch}(K_{4,\dots,4})$.Comment: 14 page

On Choosability and Paintability of Graphs

abstract: 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.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Ohba’s conjecture and beyond for generalized colorings

Let $G$ be a graph. Ohba's conjecture states that if $|V(G)|\leq 2\chi(G) +1$, then $\chi(G)=\chi^L(G)$. Noel, West, Wu and Zhu extended this result and proved that for any graph, $\chi^L(G)\leq\max\{\chi(G),\left\lceil(|V(G)+\chi(G)-1)/3\right\rceil\}$. Ohba, Kierstead and Noel proved that this bound is sharp for the ordinary chromatic number. In this work we prove that both results hold for generalized colorings as well, and find examples that prove the sharpness of the second one for the acyclic and star chromatic numbers