Euler’s Formula for General Graph Embeddings

Publication Date

1-1-2025

Document Type

Conference Proceeding

Publication Title

Springer Proceedings in Mathematics and Statistics

Volume

489

DOI

10.1007/978-3-031-83864-4_14

First Page

189

Last Page

195

Abstract

Consider an embedding of a graph G(v,e) with v≥1 vertices and e≥0 edges into a closed surface s, with r resulting regions. If G is connected and every region is a 2-cell (a so-called 2-cell embedding), Euler’s formula is the relation v−e+r=χ(s), where χ(s) denotes the Euler characteristic of s. Here we give a generalization of Euler’s formula which applies to any embedding (2-cell or not) of any graph (connected or not) into any surface (orientable or not), with several interesting corollaries. One rather striking corollary is the converse of Euler’s formula itself: If an embedding of a graph G(v,e) into a closed surface s merely has r=e−v+χ(s) regions, the right number for a 2-cell embedding, it is a 2-cell embedding.

Keywords

2-cell embeddings, Euler’s formula, Graph embeddings

Department

Mathematics and Statistics

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