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A Simplified Proof for the Edge-Density of 4-Planar Graphs

Aaron Büngener

TL;DR

This work addresses the edge-density bound for 4-planar graphs, encompassing non-simple drawings with parallel edges and loops. It presents a shorter, more general proof than Ackerman's prior argument, centered on the $H^*$ configuration and a restricted face taxonomy. A key component is proving that every 0-triangle is contained in an $H^*$ region, enabling a five-step discharging that yields $|E|\le 6(n-2)$. The result tightens understanding of beyond-planar graph density and demonstrates potential for extending these techniques to higher crossing configurations.

Abstract

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its tightness and the structural complexity of the graph class. We provide a much shorter proof while at the same time relaxing the conditions on the graph and its embedding, i.e., allowing multi-edges and non-simple drawings.

A Simplified Proof for the Edge-Density of 4-Planar Graphs

TL;DR

This work addresses the edge-density bound for 4-planar graphs, encompassing non-simple drawings with parallel edges and loops. It presents a shorter, more general proof than Ackerman's prior argument, centered on the configuration and a restricted face taxonomy. A key component is proving that every 0-triangle is contained in an region, enabling a five-step discharging that yields . The result tightens understanding of beyond-planar graph density and demonstrates potential for extending these techniques to higher crossing configurations.

Abstract

A graph on vertices drawn in the plane such that each edge is crossed at most four times has at most edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its tightness and the structural complexity of the graph class. We provide a much shorter proof while at the same time relaxing the conditions on the graph and its embedding, i.e., allowing multi-edges and non-simple drawings.

Paper Structure

This paper contains 4 sections, 1 theorem, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Proof of \ref{['th:4-planar']}
  4. Open questions

Key Result

Theorem 1

Any connected 4-planar graph $G$ on $n \ge 3$ vertices has at most $6(n-2)$ edges.

Figures (2)

  • Figure 1: (Taken from bungener2024improving.) Illustrations of the defined neighborhood relations. (a) From top to bottom: The faces $f$ and $f'$ are 0-neighbors, 1-neighbors, 2-neighbors, respectively. (b) The 0-pentagon $f_2$ is the wedge-neighbor of the 1-triangle $f_0$. (c) The faces $f$ and $f'$ are vertex-neighbors.
  • Figure 2: An $H^*$ configuration. The six vertices are not necessarily distinct; see, e.g., \ref{['fig:hexagon-b']}.

Theorems & Definitions (1)

  • Theorem 1