5 problems

Let $k$-triangulations be the maximal graphs on $[n]$ with no $(k+1)$-crossing, and let the generic bar-and-joint rigidity matroid of dimension $2k$ be the matroid obtained from ge…

No-common-edge shortest paths conjecture. 1. A shortest path in $G_{n,k}$ between $T$ and $T'$ never flips a common edge of $T$ and $T'$. 2. If $f$ is a $k$-relevant edge of $T$,…

Diameter monotonicity conjecture. The diameter satisfies

Non-flipping common contacts conjecture. The shortest path in the flip graph $G({\mathcal S})$ between two pseudoline arrangements $\Lambda$ and $\Lambda'$ supported by…

Rigidity conjecture. Every $k$-triangulation is minimally generically rigid in dimension $2k$.