CoEulerian graphs

TL;DR

This paper introduces a measure of 'Eulerianness' for directed graphs, defines coEulerian graphs with maximal Laplacian lattice size, and improves chip-firing termination algorithms for these graphs, showing NP-completeness in general cases.

Contribution

It defines coEulerian graphs based on Laplacian lattice size and provides a linear-time algorithm for chip-firing termination in these graphs, advancing understanding of graph Eulerian properties.

Findings

CoEulerian graphs have maximal Laplacian lattice size.

Linear-time algorithm for chip-firing termination on coEulerian graphs.

NP-completeness of chip-firing termination in general directed multigraphs.

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 Bjorner, Lovasz, and Shor in 1991, who asked for "a characterization of those digraphs and initial chip configurations that guarantee finite termination." Bjorner and Lovasz 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 multigraphs.