Constructing a Family of 4-Critical Planar Graphs with High Edge-Density

TL;DR

This paper constructs a new family of 4-critical planar graphs with high edge density, improving previous bounds and conjecturing maximality in edge density for such graphs.

Contribution

It introduces a novel family of 4-critical planar graphs with a higher edge density than previously known, advancing understanding of graph coloring constraints.

Findings

Constructed 4-critical planar graphs with $rac{7n-13}{3}$ edges

Improved the upper bound for edge density in 4-critical planar graphs

Conjectured this density is the maximum possible for such graphs

Abstract

A graph $G=(V,E)$ is a $k$-critical graph if $G$ is not $(k -1)$-colorable but $G-e$ is $(k-1)$-colorable for every $e\in E(G)$. In this paper, we construct a family of 4-critical planar graphs with $n$ vertices and $\frac{7n-13}{3}$ edges. As a consequence, this improved the bound for the maximum edge density obtained by Abbott and Zhou. We conjecture that this is the largest edge density for a 4-critical planar graph.