• 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}
}
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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References

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The edge density of 4-critical planar graphs

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.

On 4-critical planar graphs with high edge density

If e = v i w i+1 (v i ∈ V 2 ), let C 1 = {x 1

    If e = v i v i+1 (i is odd), let C 1 = {x 1 , y 1 } ∪ W , C 2 = {x 3 , y 2 } ∪

      Über eine Konstruktion nicht n-färbbarer Graphen

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      From the above coloring schedules

        If e = v i w i (v i ∈ V 2 ), let C 1 = {x 1

          If e = v i−1 w i (v i−1 ∈ V 1 ), let C 1 = {x 1 , y 1 } ∪ U 2 ∪ V 2 , C 2 = {x 3 , y 2 } ∪

            If e = v i v i+1 (i is even), let C 1 = {x 1 , y 1 } ∪ W , C 2 = {x 3 , y 2 } ∪