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

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.

💡 Research Summary

This paper establishes a formal combinatorial analogue of Euler’s formula for infinite, planar, regular triangular mesh graphs where every vertex has degree r > 6. For finite planar graphs, Euler’s formula v - e + f = 1 is a fundamental topological invariant. However, directly counting vertices, edges, and faces in an infinite graph yields divergent series. The authors address this by defining formal “counts” for the infinite mesh using Euler summation applied to sequences generated by an iterative geometric process.

The core methodology involves starting with an arbitrary finite, convex “combinatorial disk” G0 within the infinite mesh M. This disk is a simply-connected subgraph bounded by a cycle. An expansion operator T is defined, which, when applied to a subgraph G, adds all vertices and edges of any triangular face in M that shares at least one vertex with G. By iteratively applying this operator, one generates a sequence of growing convex disks: G_{n+1} = T(G_n). The study then focuses on the sequences v_n, e_n, f_n, which count the new vertices, edges, and faces added at the n-th expansion step.

Key technical lemmas show that convexity is preserved under expansion and that the boundary vertices of any G_n have degree either 3 or 4 within G_n. This leads to recursive formulas relating the counts of these degree-3 and degree-4 boundary vertices (a_n and b_n). A pivotal result (Proposition 1) proves that the sequences (v_n), (e_n), and (f_n) each satisfy the same second-order linear recurrence relation: x_{n+2} = (r - 4)x_{n+1} - x_n. While these sequences typically diverge as n grows, the paper employs the technique of Euler summation. For a sequence defined by such a recurrence, one can consider its formal generating function, which is a rational function. The Euler sum of the divergent series Σ x_n is defined as the value of this analytically continued generating function evaluated at t=1, yielding a finite regularized value.

The main theorem (Theorem 1) demonstrates that the Euler sums of the infinite series Σ v_n, Σ e_n, and Σ f_n—denoted as v_M, e_M, and f_M respectively—are independent of the initial choice of the convex disk G0. They depend solely on the degree r of the infinite mesh and are given by the explicit formulas: v_M = -6/(r-6), e_M = -3r/(r-6), f_M = -2r/(r-6).

Remarkably, these formal quantities, which are negative rational numbers for r>6, satisfy the classical Euler formula: v_M - e_M + f_M = 1 (Corollary 1). This result provides a precise combinatorial analogue of Euler’s formula for this class of infinite planar graphs. The work elegantly bridges combinatorial geometry, graph theory, and analytic methods for series regularization, offering a novel perspective on assigning finite invariants to infinite discrete structures.