On building 4-critical plane and projective plane multiwheels from odd wheels

TL;DR

This paper constructs infinite classes of 4-critical plane and projective plane multiwheels by combining odd wheels through edge sums modulo two, expanding understanding of critical graphs on these surfaces.

Contribution

It introduces a method to generate unbounded classes of 4-critical multiwheels from odd wheels, linking them to known graph classes like Grötzsch graphs and Mycielski constructions.

Findings

Constructed unbounded classes of 4-critical multiwheels.

Connected these classes to Grötzsch graphs and Mycielski's construction.

Identified these graphs as quadrangulating the projective plane.

Abstract

We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Gr\"otzsch class, received applying Mycielski's Construction to odd cycle.