Rotational Dimension and Colin de Verdière Invariants

Definitions and mathematical statement

Write $r(G)=\operatorname{dim}_{\mathrm{rot}}(G)$ for the source's rotational dimension and $\mu(G)$ for its Colin de Verdière invariant. The supplied record names but does not define rotational dimension; its precise optimization or representation model must therefore be recovered from the report, rather than replaced by another graph dimension. Under the usual finite simple undirected graph convention, $\mu(G)$ is the maximum corank of a real symmetric matrix $M$ having exactly one negative eigenvalue, satisfying $M_{ij}<0$ on edges and $M_{ij}=0$ on distinct nonadjacent vertices, and satisfying the Strong Arnold Property: the only symmetric matrix $X$ with $MX=0$ and $X_{ij}=0$ whenever $i=j$ or $ij$ is an edge is $X=0$. The intended graph class and any degenerate-graph conventions are not specified in the record. With these qualifications, the source poses three conjectures:

1. For every graph $G$ in the intended class, $r(G)\leq\mu(G)$.
2. If $G_1\cap G_2$ is complete, then \[r(G_1\cup G_2)\leq\max\{r(G_1),r(G_2),|V(G_1)\cap V(G_2)|+1\}.\] Here the intersection and union have the respective common and combined vertex and edge sets, and the source notation $|G_1\cap G_2|$ is interpreted as the number of common vertices. No assumption that the common clique is maximal is added.
3. For every such graph $G$, \[r(G)\leq3\quad\Longleftrightarrow\quad G\text{ has no }K_5\text{ minor}.\] A $K_5$ minor means a complete graph on five vertices obtainable by vertex deletions, edge deletions, and edge contractions.

The supplied 2026-08-21 literature assessment labels the record partially solved but explicitly says the full inequalities and characterization were not verified; it reports only small-case theorems and computational evidence in the report.

Short English statement

The record asks three questions: whether rotational dimension is bounded by the Colin de Verdière invariant, whether it obeys the stated complete-intersection union bound, and whether the value at most three is characterized by excluding a $K_5$ minor. The definition of rotational dimension is missing from the supplied extract.

Sources

• Local UnsolvedMath registry, permanent ID 30000722, source problem catalogue entry, Rotational Dimension and Colin de Verdière Invariants, proposed year 2007; no individual proposer supplied.
• Supplied citation: Graph Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/16, pp. 887--944, https://doi.org/10.4171/owr/2007/16. Supplied literature label: original Oberwolfach report. The citation's displayed year 2008 differs from the DOI's year component and the proposed-year metadata, both 2007; the discrepancy is retained.