A Catalog of Facially Complete Graphs

TL;DR

This paper provides a comprehensive classification of facially complete graphs, establishing bounds on their size and connectivity, and explores coloring properties of related graph structures.

Contribution

It offers a complete catalog of facially complete graphs, introduces a novel proof for size bounds, and analyzes coloring properties of specific face configurations.

Findings

Facially complete graphs fall into seven types.

Maximum size of such graphs is bounded by a function of the largest face.

Graphs with limited 4-faces are 5-colorable after adding diagonals.

Abstract

Considering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes complete when edges are added between every two vertices that lie on a face. Here we present a complete catalog of facially complete graphs: they fall into seven types. A consequence is that if q is the size of the largest face in a plane graph G that is facially complete, then G has at most Floor[3/2 q] vertices. This bound was known, but our proof is completely different from the 1998 approach of Chen, Grigni, and Papadimitriou. Our method also yields a count of the 2-connected facially complete graphs with n vertices. We also show that if a plane graph has at most two faces of size 4 and no larger face, then the addition of both diagonals to each 4-face leads to a graph that is 5-colorable.