Theory on Structure and Coloring of Maximal Planar Graphs (I): Relationship between Structure and Coloring

TL;DR

This paper explores the structure and coloring relationships of maximal planar graphs, introducing new construction methods, classifying 4-colorings, and laying groundwork for solving the Four-Color Conjecture.

Contribution

It presents a novel construction approach linking maximal planar graphs with 4-colorings and classifies coloring modes, advancing understanding of graph structure-coloring relations.

Findings

New construction method for maximal planar graphs

Classification of 4-colorings into tree-coloring and cycle-coloring

Necessary and sufficient condition for 2-colorable even cycles

Abstract

Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable. Indeed, in order to prove Four-Color Conjecture, it clearly suffices to show that all maximal planar graphs are 4-colorable. Since this conjecture was proposed in 1852, no mathematical proofs have been invented up until now. Maybe the main reasons lie in the following three aspects in terms of maximal planar graphs: not clearing up the structures, not figuring out the coloring types, and not straightening out the relation between structure and coloring. For this, we will write a series of articles to study the structure and coloring theory of maximal planar graphs systematically. This is our first article, which focuses mainly on the structure and coloring relations. First, we introduce a new way to construct maximal planar graphs. The advantage of this method is that it establishes an immediate relation with 4-colorings, and reveals how a given maximal planar graph is generated. Second, a special class of maximal planar graphs--recursive maximal planar graphs is researched in depth, which lays a foundation for solving the uniquely 4-colorable planar graphs conjecture(see subsequent articles). Third, we discover an important mode for classifying 4-colorings: tree-coloring and cycle-coloring, which runs through the whole series of articles. Furthermore, this mode is applied to the research on an arbitrary 4-coloring and its corresponding structure of unions of two and three bicolored subgraphs. Finally, we introduce the concepts of black-white coloring and stamen phenomenon, and find out a necessary and sufficient condition for an even cycle to be a 2-colorable cycle.