• Corpus ID: 119600091

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

@article{Tianxing2015ConstructingAF,
title={Constructing a Family of 4-Critical Planar Graphs with High Edge-Density},
author={Yao Tianxing and Zhou Guofei},
journal={arXiv: Combinatorics},
year={2015}
}
• Published 2 September 2015
• Mathematics
• arXiv: Combinatorics
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.

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Several constructions of 4-critical planar graphs are given. These provide answers to two questions of B. Grünbaum and give improved bounds for the maximum edge density of such graphs.

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