Eulerian 2-Complexes

Abstract

A famous theorem in graph theory—originating with Euler—characterizes connected even-degree graphs as (1) those graphs that admit an Euler tour, and (2) those connected graphs that decompose as a face-disjoint union of cycles. We explore a 2-dimensional generalization of this theorem, with graphs (i.e., 1-complexes) replaced by 2-complexes. This entails an interesting generalization of cycles, and the introduction of the notion of a “2-dimensional Euler tour.”