A better lower bound on average degree of k-list-critical graphs

We improve the best known bounds on average degree of $k$-list-critical graphs for $k \ge 6$. Specifically, for $k \ge 7$ we show that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for online $k$-list-critical graphs.