Edge-decompositions of $O(m)$-edge-connected graphs into isomorphic copies of a fixed tree of size $m$

TL;DR

This paper proves that highly edge-connected graphs with certain degree conditions can be decomposed into isomorphic copies of a fixed tree, improving previous bounds on edge-connectivity requirements.

Contribution

It establishes new sufficient conditions for edge-decompositions into trees, reducing the previously known factorial bounds on edge-connectivity.

Findings

Edge-decomposition into trees for graphs with high edge-connectivity

Minimum degree condition can be removed for graphs with large girth

Improves previous factorial bounds on edge-connectivity

Abstract

In this paper, we show that every $O(m)$-edge-connected simple graph $G$ of size divisible by $m$ with minimum degree at least $2^{O(m)}$ has an edge-decomposition into isomorphic copies of any given tree $T$ of size $m$. Moreover, the minimum degree condition can be dropped for graphs $G$ with girth greater than the diameter of $T$. These results improve two results due to Bensmail, Harutyunyan, Le, Merker, and Thomass\'e (2017) and Merker (2017) who gave a factorial upper bound on the necessary edge-connectivity.