Eulerian edge refinements, geodesics, billiards and sphere coloring

TL;DR

This paper explores how to transform certain finite graphs into Eulerian forms through edge refinements, enabling the construction of ergodic geodesic flows and billiards with applications in graph theory and dynamical systems.

Contribution

It provides explicit methods for making 2-graphs Eulerian via edge refinements and constructs ergodic geodesic flows and billiards on these graphs, linking graph properties to dynamical behavior.

Findings

Every 2-graph can be made Eulerian through explicit edge refinements.

2-balls can be Eulerian if and only if boundary length is divisible by 3.

Ergodic Eulerian graphs exhibit unique geodesic connections between vertices.

Abstract

A finite simple graph is called a 2-graph if all of its unit spheres S(x) are cyclic graphs of length 4 or larger. A 2-graph G is Eulerian if all vertex degrees of G are even. An edge refinement of a graph splits an edge (a,b) to two edges (a,c),(c,b) and connects the newly added vertex c to the intersection of S(a) with S(b). Theorem I assures that every 2-graph can be rendered Eulerian by successive edge refinements. The construction is explicit using geodesic cutting. After the refinement, we have an Eulerian 2-graph that carries a natural geodesic flow. We construct some ergodic ones. A 2-graph with boundary is finite simple graph for which every unit sphere is either a path graph of length n larger than 1 or a cyclic graph of length larger than 3. 2-balls are special 2-graphs are simply connected with a circular boundary. Theorem II tells that every 2-ball can be edge refined using interior edges to become Eulerian if and only if its boundary has length divisible by 3. A billiard map is defined already if all interior vertices have even degree. We will construct some ergodic billiards in 2-balls, where the geodesics bouncing off at the boundary symmetrically and which visit every interior edge exactly once. A consequence of Theorem II is that an Eulerian billiard which is ergodic must have a boundary length that is divisible by 3. We also construct other 2-graphs like tori with ergodic geodesic flows. This clashes with experience in the continuum, where tori have periodic points minimizing the length in homology classes of paths. Ergodic Eulerian 2-graphs or billiards are exciting because they satisfy a Hopf-Rynov result: there exists a geodesic connection between any two vertices. We get so unique canonical metric associated to any ergodic Eulerian graph. It is non-local in the sense that two adjacent vertices can have large distance.