Boxicity of higher-dimensional Kneser graphs

Determine the boxicity of the Kneser graphs \(\mathrm{KG}(n,k)\) for all \(k\geq 3\), including the unresolved case \(\mathrm{KG}(7,3)\).

Background

The paper proves that $\boxi(\mathrm{KG}(n,2))=n-2$, thereby resolving the previously conjectured case k=2k=2. The authors state that the corresponding determination for k≥3k\geq 3 remains unresolved and specifically identify $\boxi(\mathrm{KG}(7,3))$ as an open test case.

The paper notes two possible patterns consistent with the established k=2k=2 result: $\boxi(\mathrm{KG}(n,k))=\chi(\mathrm{KG}(n,k))=n-2k+2$, or alternatively $\boxi(\mathrm{KG}(n,k))=n-k$. These are presented as possibilities rather than established conjectures.

References

While this question is resolved in this paper for the case $k=2$, the general case remains open.

\begin{problem} Determine the boxicity of Kneser graphs $\KG(n,k)$ with $k\geq 3$. \end{problem} Already establishing $\boxi(\KG(7,3))$ is open.

— On the Boxicity of Line Graphs and of Their Complements  (2501.05049 - Caoduro et al., 9 Jan 2025) in Section 7, Conclusion, immediately before Problem 1; page number unavailable in the source