# The list chromatic number of graphs with small clique number

## TL;DR

This paper establishes new upper bounds on the list chromatic number of graphs with small clique number, particularly for triangle-free graphs and graphs with larger clique restrictions, advancing understanding of graph coloring complexities.

## Contribution

It provides improved bounds on the list chromatic number for triangle-free graphs and $K_r$-free graphs, with new proofs and asymptotic results.

## Key findings

- Triangle-free graphs with maximum degree Δ have list chromatic number at most (1+o(1))Δ/ln Δ.
- For any r ≥ 4, K_r-free graphs have list chromatic number at most 200r (Δ ln ln Δ)/ln Δ.
- Results match bounds known for graphs with girth at least 5.

## Abstract

We prove that every triangle-free graph with maximum degree $\Delta$ has list chromatic number at most $(1+o(1))\frac{\Delta}{\ln \Delta}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r\geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{\Delta\ln\ln\Delta}{\ln\Delta}$.