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 [Formula presented] (k−1+ [Formula presented] )n edges where c=(k−3)( [Formula presented] − [Formula presented] ). This improves the bound established by Kostochka and Stiebitz [11]. 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.