Choosability of Graph Powers ★★

Author(s): Noel

\begin{question}[Noel, 2013] Does there exist a function $f(k)=o(k^2)$ such that for every graph $G$, \[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\] \end{question}

Keywords: choosability; chromatic number; list coloring; square of a graph

Bounding the on-line choice number in terms of the choice number ★★

Author(s): Zhu

\begin{question} Are there graphs for which $\text{ch}^{\text{OL}}-\text{ch}$ is arbitrarily large? \end{question}

Keywords: choosability; list coloring; on-line choosability

Choice number of complete multipartite graphs with parts of size 4


\begin{question} What is the choice number of $K_{4*k}$ for general $k$? \end{question}

Keywords: choosability; complete multipartite graph; list coloring

Choice Number of k-Chromatic Graphs of Bounded Order ★★

Author(s): Noel

\begin{conjecture} If $G$ is a $k$-chromatic graph on at most $mk$ vertices, then $\text{ch}(G)\leq \text{ch}(K_{m*k})$. \end{conjecture}

Keywords: choosability; complete multipartite graph; list coloring

Circular choosability of planar graphs

Author(s): Mohar

Let $G = (V, E)$ be a graph. If $p$ and $q$ are two integers, a $(p,q)$-colouring of $G$ is a function $c$ from $V$ to $\{0,\dots,p-1\}$ such that $q \le |c(u)-c(v)| \le p-q$ for each edge $uv\in E$. Given a list assignment $L$ of $G$, i.e.~a mapping that assigns to every vertex $v$ a set of non-negative integers, an $L$-colouring of $G$ is a mapping $c : V \to N$ such that $c(v)\in L(v)$ for every $v\in V$. A list assignment $L$ is a $t$-$(p,q)$-list-assignment if $L(v) \subseteq \{0,\dots,p-1\}$ and $|L(v)| \ge tq$ for each vertex $v \in V$ . Given such a list assignment $L$, the graph G is $(p,q)$-$L$-colourable if there exists a $(p,q)$-$L$-colouring $c$, i.e. $c$ is both a $(p,q)$-colouring and an $L$-colouring. For any real number $t \ge 1$, the graph $G$ is $t$-$(p,q)$-choosable if it is $(p,q)$-$L$-colourable for every $t$-$(p,q)$-list-assignment $L$. Last, $G$ is circularly $t$-choosable if it is $t$-$(p,q)$-choosable for any $p$, $q$. The circular choosability (or circular list chromatic number or circular choice number) of G is $$cch(G) := \inf\{t \ge 1 : G \text{ is circularly $t$-choosable}\}.$$

\begin{problem} What is the best upper bound on circular choosability for planar graphs? \end{problem}

Keywords: choosability; circular colouring; planar graphs

Ohba's Conjecture ★★

Author(s): Ohba

\begin{conjecture} If $|V(G)|\leq 2\chi(G)+1$, then $\chi_\ell(G)=\chi(G)$. \end{conjecture}

Keywords: choosability; chromatic number; complete multipartite graph; list coloring

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