Edge-partitioning a graph into paths: beyond the Bar\'at-Thomassen conjecture

TL;DR

This paper presents an alternative proof for partitioning highly edge-connected graphs into paths of a given length, reducing the connectivity requirements compared to previous results, especially for Eulerian graphs.

Contribution

It introduces a new proof method that lowers the edge-connectivity threshold for path partitions, notably achieving results with only 4-edge connectivity for Eulerian graphs.

Findings

Partitioning into paths is possible with 24-edge-connected graphs.

Eulerian graphs require only 4-edge connectivity for such partitions.

The approach broadens understanding of graph decompositions into paths.

Abstract

The Bar\'at-Thomassen conjecture asserts that there is a function $f$ such that for every fixed tree $T$ with $t$ edges, every graph which is $f(t)$-edge-connected with its number of edges divisible by $t$ has a partition of its edges into copies of $T$. This has been proved in the case of paths of length $2^k$ by Thomassen, and recently shown to be true for all paths by Botler, Mota, Oshiro and Wakabayashi. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function $f$ such that every $24$-edge-connected graph with minimum degree $f(t)$ has an edge-partition into paths of length $t$ whenever $t$ divides the number of edges. We also show that $24$ can be dropped to $4$ when the graph is eulerian.