Abstract
Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem states that $G$ has a $\Delta$-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, Král', 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.
Highlights
Brooks’ theorem is one of the fundamental results in graph coloring
For every connected graph G, it says that G has a ∆-coloring unless G is a complete graph K∆+1 or an odd cycle
Rubin, and Taylor [5] and Borodin [3] strengthened this Brooks’ analogue to the following result, where a Gallai tree is a connected graph in which each block is a complete graph or an odd cycle
Summary
Brooks’ theorem is one of the fundamental results in graph coloring. For every connected graph G, it says that G has a ∆-coloring unless G is a complete graph K∆+1 or an odd cycle. When we seek to prove coloring results by induction, we often want to color a subgraph H where different vertices have different lists of allowable colors (those not already used on their neighbors in the coloring of G − H). Vizing [15] and, independently, Erdos, Rubin, and Taylor [5] extended Brooks’ theorem to list coloring. They proved an analogue of Brooks’ theorem when each vertex v has ∆ allowable colors (possibly different colors for different vertices). Rubin, and Taylor [5] and Borodin [3] strengthened this Brooks’ analogue to the following result, where a Gallai tree is a connected graph in which each block is a complete graph or an odd cycle
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