CoEulerian graphs

Matthew Farrell, Lionel Levine

Abstract: We suggest a measure of “Eulerianness” of a finite directed graph and define a class of “coEulerian” graphs. These are the graphs whose Laplacian lattice is as large as possible. As an application, we address a question in chip-firing posed by Bj¨orner, Lov´asz, and Shor in 1991, who asked for “a characterization of those digraphs and initial chip configurations that guarantee finite termination.” Björner and Lovász gave an exponential time algorithm in 1992. We show that this can be improved to linear time if the graph is coEulerian, and that the problem is NP-complete for general directed multi-graphs.