choosability


Choosability of Graph Powers ★★

Author(s): Noel

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

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

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

Keywords: choosability; list coloring; on-line choosability

Choice number of complete multipartite graphs with parts of size 4

Author(s):

Question   What is the choice number of $ K_{4*k} $ for general $ k $?

Keywords: choosability; complete multipartite graph; list coloring

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

Author(s): Noel

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

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 $$cch(G) := \inf\{t \ge 1 : G \text{ is circularly $t$-choosable}\}.$$

Problem   What is the best upper bound on circular choosability for planar graphs?

Keywords: choosability; circular colouring; planar graphs

Ohba's Conjecture ★★

Author(s): Ohba

Conjecture   If $ |V(G)|\leq 2\chi(G)+1 $, then $ \chi_\ell(G)=\chi(G) $.

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

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