List Colouring When The Chromatic Number Is Close To the Order Of The Graph

Abstract

Ohba has conjectured [7] that if G has 2χ(G)+1 or fewer vertices then the list chromatic number and chromatic number of G are equal. In this short note we prove the weaker version of the conjecture obtained by replacing 2χ(G) + 1 by 5 3 χ(G) − 4 3 .

On the choice number of complete multipartite graphs with part size four

“…The following lemma was proved independently by Kierstead [7], and by Reed and Sudakov [11,12], and named by Rabern. Pot Lemma).…”

A Proof of a Conjecture of Ohba

“…For one, we show that for any minimal counterexample G, if G is not L-colorable, then the total number of colors in the union of the lists of L is at most 2k. This upper bound on the number of colors, foreshadowed in earlier work of Kierstead [12] and Reed and Sudakov [24,25], is crucial in that it implies the existence of colors that appear in the lists of many vertices, which our approach requires.…”

“…If any of the following are true, then G is chromatic-choosable. [25]); (c) |V (G)| ≤ (2 − ε)k − n 0 (ε) for some function n 0 of ε ∈ (0, 1). (Reed and Sudakov [24] [16]; Shen et al [26] proved the result for α ≤ 3).…”

Three topics in online list coloring

“…The coloring algorithm, which we call Painter, must choose an independent set of marked vertices to receive that color. Colored Ohba [11] conjectured that G is chromatic-choosable when |V (G)| ≤ 2χ(G) + 1; after partial results in [8,11,12,14], this was proved by Noel, Reed, and Wu [10]. Various researchers (see [7]) observed that the complete multipartite graph K 2,...,2,3 is chromatic-choosable but not chromatic-paintable, so the paintability analogue is slightly different: Conjecture 1.2.…”