Absence of $1$-Nearly Platonic Graphs

TL;DR

This paper proves the non-existence of 1-nearly platonic graphs, which are planar graphs with all but one face having the same length, extending previous results that only covered 2-connected cases.

Contribution

It establishes that no 1-nearly platonic graphs exist, filling a gap in the classification of nearly platonic graphs.

Findings

No 1-nearly platonic graphs exist.

Extends previous non-existence results to all cases.

Provides a complete characterization of nearly platonic graphs.

Abstract

A $t$-nearly platonic graph is a finite, connected, regular, simple and planar graph in which all but exactly $t$ numbers of its faces have the same length. It is proved that there is no 2-connected $1$-nearly platonic graph. In this paper, we prove that there is no $1$-nearly platonic graph.