Packing Trees into 1-planar Graphs

TL;DR

This paper investigates the 1-planar packing problem for multiple graphs, especially trees, establishing existence and non-existence results, and providing bounds on crossings for certain graph combinations.

Contribution

It introduces the 1-planar packing problem for trees, proves new existence and non-existence results, and bounds the crossings for specific graph configurations.

Findings

Certain triples of trees do not admit 1-planar packings.

Two paths and a special caterpillar always admit a 1-planar packing.

Three paths or cycles can be packed with bounded crossings.

Abstract

We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$ is a tree and $k=3$. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with $n \geq 12$ vertices admits a 1-planar packing, while such a packing does not exist if $n \leq 10$.