Eulerian 2-Complexes

TL;DR

The paper generalizes Euler's equivalence from graphs to strongly connected 2-complexes by introducing circlets as minimal even 2-complexes and Euler covers as maps from a connected surface that cover each 2-cell exactly once. It proves the equivalence: a 2-complex $K$ is even iff it is a face-disjoint union of circlets iff $K$ has an Euler cover, with explicit constructions showing each circlet admits an Euler cover. A key contribution is the decomposition of any even 2-complex into circlets and the constructive splicing of individual Euler covers into a global cover. The results yield concrete decompositions for familiar 2-skeletons (e.g., of simplices and Platonic solids) and lay groundwork for higher-dimensional generalizations, while noting caveats related to manifold structure in dimensions $>2$.

Abstract

It is shown that Euler's theorem for graphs can be generalized for 2-complexes. Two notions that generalize cycle and Eulerian tour are introduced (``circlet'' and ``Eulerian cover''), and we show that for a strongly-connected, pure 2-complex, the following are equivalent: (i) each edge meets a positive even number of 2-cells (faces), (ii) the complex can be decomposed as the face-disjoint union of circlets, and (iii) the complex has an Eulerian cover. A number of examples are provided.

Figures (15)

• Figure 1: Left: A network of bridges. Center: The graph that models the network. Right: A graph that meets the conditions of the Eulerian equivalence. arXiv copy, 2023
• Figure 2: A 2-complex $K$.
• Figure 3: An Euler tour $\varphi\colon C_{10}\to G$ in the graph $G= K_5$. Here $V(G)=\{1,2, 3,4,5\}$, and each vertex $x$ of $C_{10}$ is labeled with $\varphi(x)$. The solid and dashed paths in $C_{10}$ map to two edge-disjoint cycles in $G$, as illustrated.
• Figure 4: An Euler cover $\varphi\colon M\to \Delta_5^{2}$ of the 2-skeleton of the 5-dimensional simplex, which has six vertices $1,2,3,4,5,6$, fifteen edges and twenty triangular faces. Here $M$ is a triangulated sphere, with each vertex $x$ labeled by $\varphi(x)$. The three shaded areas of $M$ map to three face-disjoint tetrahedra. The white areas of $M$ (including the unbounded face) map to an octahedron sharing no face with the three tetrahedra.
• Figure 5: Examples and non-examples of circlets. The pinched (or zipped) spheres $K$ and $K'$ are circlets, but the zipped sphere $K"$ is not.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Proposition 1
• proof
• proof