A Formal Analogue of Euler's Formula for Infinite Planar Regular Graphs

TL;DR

This paper develops a formal framework for understanding the relationships between vertices, edges, and faces in infinite planar regular graphs, extending Euler's formula to an infinite setting using combinatorial and summation techniques.

Contribution

It introduces a formal method to define and verify Euler's formula for infinite planar regular graphs of degree greater than six, using Euler summation of combinatorial sequences.

Findings

Formal quantities satisfy Euler's formula for infinite graphs

Extension of Euler's formula to infinite regular triangulations

Uses combinatorial and summation techniques for proof

Abstract

We present a formal version of the numbers of vertices, edges, and faces for infinite planar regular triangular meshes of degree r>6. These numbers are defined via Euler summation of sequences obtained from iterated expansions of a convex combinatorial disk. We prove that these formal quantities satisfy the classical Euler formula, providing a combinatorial analogue of Euler's formula for infinite planar graphs.