Beyond Degree Choosability
Daniel W. Cranston, Landon Rabern

TL;DR
This paper explores advanced graph coloring concepts, extending Brooks' theorem and analyzing Alon--Tarsi orientations, to characterize specific orientations in connected graphs with a focus on a designated vertex.
Contribution
It provides a characterization of pairs (G, x) where G lacks certain Alon--Tarsi orientations, especially in 2-connected graphs, extending previous degree-choosability results.
Findings
Characterization of pairs (G, x) with no suitable Alon--Tarsi orientation.
Simplified description for 2-connected graphs.
Extension of degree-choosability and Alon--Tarsi orientation theory.
Abstract
Let be a connected graph with maximum degree . Brooks' theorem states that has a -coloring unless is a complete graph or an odd cycle. A graph is \emph{degree-choosable} if can be properly colored from its lists whenever each vertex gets a list of colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph is degree-choosable unless each block of is a complete graph or an odd cycle; such a graph is a \emph{Gallai tree}. This degree-choosability result was further strengthened to Alon--Tarsi orientations; these are orientations of 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 is \emph{degree-AT} if has an Alon--Tarsi orientation in which each vertex has indegree at…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
