# Polygonal graph

Polygonal graphs are a combinatorial structures that generalize the idea of a regular polyhedron. Like abstract polyhedra they don't intrinsically have spacial positioning but they have several key differences from abstract regular polyhedra.

## Motivation

Abstract regular polygons can be seen as connected graphs with vertices of degree 2. Polygonal graphs attempt to generalize this idea to regular polyhedra. They thus start with a skeleton graph and add faces in the form of cycles on top.

## Definition

A polygonal graph is pair, (G , E ) consisting of a graph G  and a set ${\displaystyle E}$ of cycles in G  called "polygons", such that:

• G  is regular (i.e. every vertex has the same degree)
• G  is connected.
• G  has girth m  ≥ 3.
• Every polygon in E  has size m .
• Every subgraph in G  which is isomorphic to the star graph ${\displaystyle S_{2}}$ is a subgraph of exactly one polygon in E .

## Strict polygonal graphs

A polygonal graph is strict when its polygons consist of all the cycles of size m  on its graph.

## Comparison to related concepts

### Abstract regular polyhedra

The cube (left) is a polygonal graph. Its Petrial (right) is not because its faces have size 6 but the underlying graph, Q n , has girth 4. Both are abstract polytopes.

While there is wide overlap between polygonal graphs and abstract regular polyhedra there are some key differences:

• Abstract regular polyhedra allow faces to share more than 1 consecutive edges. For example dihedra are abstract regular polyhedra but not polygonal graphs. All such cases are considered degenerate.
• Abstract regular polyhedra allow digonal faces. For example hosohedra are abstract regular polyhedra but not polygonal graphs. All such cases are also considered degenerate.
• Polygonal graphs allow polyhedra which are not flag transitive so as they are equivelar. For example the Heawood map is a polygonal graph, but it is not considered regular as an abstract polytope.
• Polygonal graphs must have faces with size equal to the girth of the skeleton. Abstract regular polyhedra do not have this restriction.
• Polygonal graphs potentially allow more than two cycles to meet at an edge. Abstract polyhedra require dyadicity.

## Bibliography

• Perkel, Manley (1979). "Bounding the valancy of polygonal graphs" (PDF). Canadian Journal of Math. 31 (6): 1307–1321.
• Perkel, Manley (1977). On finite groups acting on polygonal graphs (PhD thesis). University of Michigan.