Negative type of 3-subdivisions of graphs with a K_{2,3} minor

Prove that for every graph \(G\) containing \(K_{2,3}\) as a minor, the shortest-path metric of the 3-subdivision of \(G\) does not have negative type.

Background

The paper proves that if a graph contains K2,3K_{2,3} as a minor, then the shortest-path metric of its 180-subdivision does not have negative type. It also shows that the shortest-path metric of the 2-subdivision of K4K_4 is ℓ1\ell_1-embeddable, and therefore has negative type.

These results motivate the conjecture that subdivision length 3 already suffices to force failure of negative type for every graph containing a K2,3K_{2,3} minor. Establishing this would substantially improve the subdivision bound obtained from the paper’s main theorem.

References

In light of the above theorems, we propose the following conjecture. There is no graph $G$ with a $K_{2,3}$ minor where the shortest distance metric of the $3$-subdivision of $G$ has negative type.

— Metric graphs of negative type  (2501.07098 - Campbell et al., 13 Jan 2025) in Subsection “Subdivisions of graphs,” immediately following the proposition on the 2-subdivision of \(K_4\)