Removing coning hypotheses in dimensions two and three

Determine whether hypotheses (a) and (b) in Theorem 5.6 can be removed when the dimension is \(d=2\) or \(d=3\), so that the cone edges incident with any two technicolour vertices of a \(k\)-fold \(\mathcal{R}_d\)-circuit necessarily belong to different parts of the principal partition.

Background

Theorem 5.6 gives sufficient conditions ensuring that the cone edges $vu$ and $vx$ associated with two technicolour vertices lie in different parts of the principal partition of the coned circuit. The paper gives a counterexample showing that these conditions cannot generally be removed for $d\geq 4$.

The authors prove that the conditions can be removed in dimension one, but leave dimensions two and three unresolved. They also note that the only known counterexample in dimensions at least four is flexible, suggesting a further unresolved possibility for rigid graphs.