Packing Steiner Trees

TL;DR

This paper improves the bounds on the number of edge-disjoint Steiner trees guaranteed in a graph under certain connectivity conditions, advancing the understanding of Kriesell's conjecture.

Contribution

The paper presents a new bound of 5k+4 for the number of edge-disjoint T-Steiner trees, improving previous bounds and making progress on Kriesell's conjecture.

Findings

Established a bound of 5k+4 for edge-disjoint Steiner trees

Improved upon previous bounds of 6.5k and 24k

Contributed to the progress on Kriesell's conjecture

Abstract

Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided that every edge-cut of $G$ that separates $T$ has size $\ge 2k$. When $T=V(G)$ a $T$-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when $2k$ is replaced by $24k$, and recently West and Wu have lowered this value to $6.5k$. Our main result makes a further improvement to $5k+4$.