Beyond Degree Choosability

Let G be a connected graph with maximum degree ∆. Brooks’ theorem states that G has a ∆-coloring unless G is a complete graph or an odd cycle. A graph G is degree-choosable if G can be properly colored from its lists whenever each vertex v gets a list of d(v) colors. In the context of list coloring, Brooks’ theorem can be strengthened to the following. Every connected graph G is degree-choosable unless each block of G is a complete graph or an odd cycle; such a graph G is a Gallai tree. This degree-choosability result was further strengthened to Alon–Tarsi orientations; these are orientations of G in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph G is degree-AT if G has an Alon–Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if G is degree-AT, then G is also degree-choosable. Hladký, Krá ’ l, and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs (G, x) where G is a connected graph and x is some specified vertex in V (G). We characterize pairs such that G has no Alon–Tarsi orientation in which each vertex has indegree at least 1 and x has indegree at least 2. When G is 2-connected, the characterization is simple to state.