Beyond Ohba’s Conjecture: A bound on the choice number ofk-chromatic graphs withnvertices

Abstract

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

“…Yang [17] proved ⌈ 3k 2 ⌉ ≤ ch(K 4 * k ) ≤ ⌈ 7k 4 ⌉, and Noel, West, Wu and Zhu [9] improved the upper bound to ⌈ 5k−1 3 ⌉. The main result of this paper is that (1.1) holds for s = 4.…”

A Proof of a Conjecture of Ohba

“…In general, one may ask the following: given a function f (k) > 2k + 1, what is the best bound on χ (G)for k-chromatic graphs on at most f (k) vertices? By applying the result of the current article, Noel et al [21] have solved this problem for every function f such that f (k) ≤ 3k and f (k) − k is even for all k. Their main result is the following strengthening of Ohba's Conjecture, which holds for all graphs G:…”

Minimum non-chromatic-choosable graphs with given chromatic number

“…An innovative idea introduced in [14] is to use a near acceptable L-colouring to derive a proper L-colouring of G. The definition of near acceptable L-colouring in this paper is slightly different from that given in [14]. Neverthless, we shall show that the same conclusion holds: Lemma 6.1 If G has a near acceptable L-colouring f , then G has an L-colouring.…”

“…Ohba [16] conjectured that for any positive integer k, k-colourable graphs with at most 2k +1 vertices are k-choosable. This conjecture has been studied in many papers [10,12,[14][15][16][17][19][20][21], and was confirmed by Noel, Reed and Wu [15]: Theorem 1.1 Every k-colourable graph with at most 2k + 1 vertices is k-choosable.…”

“…Since it was proved in [14] that k-chromatic graphs with at most 2k + 1 vertices are k-choosable, we may assume that G has exactly 2k + 2 vertices.…”