Independent sets in the union of two Hamiltonian cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by f(n, k): the maximal number of Hamiltonian cycles on an n element set, such that no two cycles share a common independent set of size more than k. We shall mainly be interested in the behavior of f(n, k) when k is a linear function of n, namely k = cn. We show a threshold phenomenon: there exists a constant e t such that for c < c(t), f (n, cn) is bounded by a constant depending only on c and not on n, and for c(t) < c, f (n, cn) is exponentially large in n (n -> infinity). We prove that 0.26 < c(t) < 0.36, but the exact value of c(t) is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of K-4 subgraphs. A corollary of this lemma is that if a graph G on n > 12 vertices is the union of two Hamiltonian cycles and alpha(G) = n/4, then V (G) can be covered by vertex-disjoint K-4 subgraphs.