Edge-decomposition of graphs into copies of a tree with four edges

TL;DR

This paper investigates edge-decompositions of highly connected graphs into copies of a specific four-edge tree, proving the conjecture for bipartite graphs and providing the first such result for a non-path, non-star tree.

Contribution

It proves the Barát-Thomassen conjecture for bipartite graphs and establishes a decomposition into a particular four-edge tree in highly connected graphs.

Findings

Proved the conjecture for bipartite graphs.

Established a 191-edge-connected graph with a Y-decomposition.

First result for a non-path, non-star tree decomposition.

Abstract

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar\'at and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a 191-edge-connected graph of size divisible by 4, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star.