Which Johnson graphs have Catalan clique-covering number?
TheoremDB P11526 · graph theory
Inferred from the community field; finer classification has not been reviewed.
Problem statement
Let be the Johnson graph whose vertices are the k-subsets of a 2k-element set, with two vertices adjacent when their intersection has size k-1. Let theta(G) be the minimum number of cliques whose vertex sets cover V(G). For which positive integers k does theta(J(2k,k)) equal the k-th Catalan number C_k? In particular, does equality hold for every even k?
Lean 4
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Background
Additional formulation recorded by the source
\text{Characterize }{k\ge1:\theta(J(2k,k))=C_k};\quad \forall k\equiv0\pmod2,\ \theta(J(2k,k))=C_k,?
Progress & source assessment
Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
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