Packing $T$-connectors in graphs needs more connectivity

TL;DR

This paper disproves a conjecture that highly edge-connected graphs always contain multiple edge-disjoint $T$-connectors, by providing counterexamples for certain connectivity levels.

Contribution

The authors construct infinite counterexamples to a conjecture on $T$-connectors in graphs, challenging previous assumptions about connectivity requirements.

Findings

Counterexamples for $k=1$ and even $k$ disprove the conjecture.

High edge-connectivity does not guarantee multiple $T$-connectors.

The conjecture by West and Wu is false in general.

Abstract

Strengthening the classical concept of Steiner trees, West and Wu [J. Combin. Theory Ser. B 102 (2012), 186--205] introduced the notion of a $T$-connector in a graph $G$ with a set $T$ of terminals. They conjectured that if the set $T$ is $3k$-edge-connected in $G$, then $G$ contains $k$ edge-disjoint $T$-connectors. We disprove this conjecture by constructing infinitely many counterexamples for $k=1$ and for each even $k$.