Generalizations of the Tree Packing Conjecture

TL;DR

This paper explores generalizations of the Gyárfás tree packing conjecture, proving packing results for trees into k-chromatic graphs under specific conditions, such as most being stars.

Contribution

It extends the conjecture by proving packings into k-chromatic graphs when all but three trees are stars, and explores additional generalizations.

Findings

Packings into k-chromatic graphs when all but three trees are stars

Validation of the conjecture in new graph classes

Extension of tree packing results to broader scenarios

Abstract

The Gy\'arf\'as tree packing conjecture asserts that any set of trees with $2,3, ..., k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into $k$-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any $k$-chromatic graph. We also consider several other generalizations of the conjecture.