# Line graphs of complete hypergraphs as complement of Kneser graphs

Since the Johnson graph/triangular graph $$J(n,2)$$ is the complement of the Kneser graph $$K(n,2)$$, which is also incidentally the line graph of the complete graph $$K_n$$, I thought whether the same can be said about the line graphs of the complete hypergraphs $$H_n^k$$. That is, can we say that the line graphs of the complete hypergraphs $$H_n^k$$ are isomorphic to the complements of Kneser graphs $$K(n,k)$$? The Baranyai's theorem and the Erdos-Ko-Rado theorems seem to be pointing out in a similar direction. Any hints? Thanks beforehand.

Yes, the correspondence goes the following way: The complete k-uniform Hypergraph has n vertices and the edges are given by all k-element subsets of {1,...,n}. Thus, the line graph has those k-element subsets as vertices and they are adjacent if and only if they intersect non-trivially. The complement graph, thus, has the same vertex set but the vertices are adjacent if and only if they intersect trivially. This is the definition of the Kneser graph.

• So the answer is in the affirmative right! Commented Aug 4, 2022 at 16:38