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

2025/12/16 by Jędrzejewicz, Piotr, Marciniak, Mikołaj
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.14030

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.

Citations

• Divergent Infinite Series -- Ramanujan's Initial Intuition

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