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 $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$andeverynon-complete$6$-list-criticalgraphhasaveragedegreeatleast$5 + \frac{93}{766}$. The same bounds hold for online $k$-list-critical graphs.