Catalan clique-cover equality for all even k

Determine whether the clique covering number of the Johnson graph J(2k, k) equals the kth Catalan number C_k for every even positive integer k.

Background

The paper establishes θ(J(2k, k)) = C_k for k = 1, 2, 4, 6, 8, and 16. For other even values of k, no explicit values of θ(J(2k, k)) are obtained, and the existence of covers by disjoint maximum cliques remains unresolved.

The question asks whether the Catalan lower bound is always attained for even k, extending the known cases and potentially encompassing values not covered by the power-of-two lexicode conjecture.

References

For other even values of $k$ we do not have explicit values of $\theta(J(2k, k))$, leaving open the possibility that we do have covers by disjoint maximum cliques in those cases too. Is $\theta(J(2k, k)) = C_k$ whenever $k$ is even?

— On the clique covering numbers of Johnson graphs  (2502.15019 - Jørgensen, 20 Feb 2025) in Question following the summary paragraph in Section 4, “Bounds from coding theory”