Improved lower bounds on the number of edges in list critical and online list critical graphs

We prove that every k-list-critical graph (k ≥ 7) on n ≥ k + 2 vertices has at least 1 2 ( k − 1 + k−3 (k−c)(k−1)+k−3 ) n edges where c = (k − 3) ( 1 2 − 1 (k−1)(k−2) ) . This improves the bound established by Kostochka and Stiebitz [13]. The same bound holds for online k-list-critical graphs, improving the bound established by Riasat and Schauz [16]. Both bounds follow from a more general result stating that either a graph has many edges or it has an Alon-Tarsi orientable induced subgraph satisfying a certain degree condition.