The Planar Tree Packing Theorem

TL;DR

This paper proves the conjecture that any pair of non-star trees can be packed into a planar graph without edge overlaps, providing a constructive polynomial-time algorithm for such packings.

Contribution

It establishes the planar tree packing theorem, confirming that all non-star tree pairs can be packed into a planar graph, and offers a polynomial-time algorithm for the packing.

Findings

Proved the planar tree packing conjecture for all non-star trees.

Provided a polynomial-time algorithm for constructing the packings.

Confirmed the only exception is star trees, which cannot be packed with others.

Abstract

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.