On Partitioning the Edges of 1-Plane Graphs

TL;DR

This paper studies edge colorings of optimal 1-plane graphs, showing how to partition edges into non-crossing sets with specific properties, and discusses implications for graph augmentation and drawing.

Contribution

It proves that optimal 1-plane graphs can be edge-colored so that one color forms a maximal planar subgraph and the other has bounded degree, with optimal bounds.

Findings

Blue subgraph is maximal planar

Red subgraph has maximum degree four

Red-blue coloring may not produce a red forest of bounded degree

Abstract

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red-blue edge coloring of an optimal 1-plane graph $G$ partitions the edge set of $G$ into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: $(i)$ Every optimal 1-plane graph has a red-blue edge coloring such that the blue subgraph is maximal planar while the red subgraph has vertex degree at most four; this bound on the vertex degree is worst-case optimal. $(ii)$ A red-blue edge coloring may not always induce a red forest of bounded vertex degree. Applications of these results to graph augmentation and graph drawing are also discussed.