## Abstract

A graph *G* is *k**-critical* if \(\chi (G) = k\) and every proper subgraph of *G* is \((k - 1)\)-colorable, and if *L* is a list assignment for *G*, then *G* is *L**-critical* if *G* is not *L*-colorable but every proper subgraph of *G* is. In 2014, Kostochka and Yancey proved a lower bound on the average degree of an *n*-vertex *k*-critical graph tending to \(k - \frac{2}{k - 1}\) for large *n* that is tight for infinitely many values of *n*, and they asked how their bound may be improved for graphs not containing a large clique. Answering this question, we prove that there exists some \(\varepsilon > 0\) for which the following holds. If *k* is sufficiently large and *G* is a \(K_{\omega + 1}\)-free *L*-critical graph where \(\omega \le k - \log ^{10}k\) and *L* is a list assignment for *G* such that \(|L(v)| = k - 1\) for all \(v\in V(G)\), then the average degree of *G* is at least \((1 + \varepsilon )(k - 1) - \varepsilon \omega - 1\). This result implies that for some \(\varepsilon > 0\), for every graph *G* satisfying \(\omega (G) \le \textrm{mad}(G) - \log ^{10}\textrm{mad}(G)\) where \(\omega (G)\) is the size of the largest clique in *G* and \(\textrm{mad}(G)\) is the maximum average degree of *G*, the list-chromatic number of *G* is at most .

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We thank the anonymous referees for their careful reading of this paper and their suggestions.

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T. Kelly: Partially supported by the EPSRC, Grant No. EP/N019504/1. L. Postle: Partially supported by NSERC under Discovery Grant No. 2019-04304, the Ontario Early Researcher Awards program and the Canada Research Chairs program.

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Kelly, T., Postle, L. On the Density of Critical Graphs with No Large Cliques.
*Combinatorica* **43**, 57–89 (2023). https://doi.org/10.1007/s00493-023-00007-w

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DOI: https://doi.org/10.1007/s00493-023-00007-w