Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

TL;DR

This paper proves a weaker form of the Tree Decomposition Conjecture, showing that highly edge-connected graphs can be decomposed into homomorphic copies of a fixed tree, especially for trees with diameter at most 4.

Contribution

It introduces a weaker version of the conjecture involving homomorphic copies and verifies it for all trees with diameter at most 4.

Findings

Proves a weaker version of the Tree Decomposition Conjecture.

Verifies the conjecture for all trees of diameter at most 4.

Establishes conditions under which graphs can be decomposed into homomorphic copies.

Abstract

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to $T$. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from $T$ by vertex-identifications. We call such a subgraph a homomorphic copy of $T$. This implies the Tree Decomposition Conjecture under the additional constraint that the girth of $G$ is greater than the diameter of $T$. As an application, we verify the Tree Decomposition Conjecture for all trees of diameter at most 4.