The list chromatic number of graphs with small clique number

Abstract

We prove that every triangle-free graph with maximum degree ∆ has list chromatic number at most (1 + o(1)) ∆ ln ∆ . This matches the best-known upper bound for graphs of girth at least 5. We also provide a new proof that for any r ≥ 4 every K r -free graph has list-chromatic number at most 200r ∆ ln ln ∆ ln ∆ .

The Johansson‐Molloy theorem for DP‐coloring

“…The proof of Lemma is virtually identical to that of [, Lemma 7], so we first show how to derive Lemma from Lemma (this is the new ingredient in our version of Molloy's argument).…”

“…Subsequently, Pettie and Su improved the bound to C = 4. Very recently, Molloy reduced the constant to C = 1:…”

“…The two main new ideas that allowed Molloy to dramatically simplify Johansson's proof and establish Theorem are: –a new coupon collector‐type result [, Lemma 7] with elements drawn uniformly at random from possibly distinct sets; and –the use of the entropy compression method instead of iterated applications of the Lovász Local Lemma. …”

“…Another result of Johansson asserts that if G is K r ‐free for some fixed r ≥ 4. Molloy [, Theorem 2] also gave a new short proof of this bound with explicit dependence on r . In Section 4, we extend it to DP‐coloring:…”

“…The main technical step in the proof of Theorem is Lemma . It is similar to [, Lemma 15]; however, it is necessary to modify the proof of [, Lemma 15] somewhat in order to adapt it for the DP‐coloring framework, since, in contrast to list coloring, a DP‐coloring of a graph cannot be naturally represented as a partition of its vertex set into independent subsets.…”