Binomial and Q-binomial coefficient inequalities related to the Hamiltonicity of the Kneser graphs and their Q-analogues

The Kneser graph K(n, k) has as vertices all the k-subsets of a fixed n-set and has as edges the pairs {A, B} of vertices such that A and B are disjoint. It is known that these graphs are Hamiltonian if ((n-1)(k-1)) less than or equal to ((n-k)(k)) for n greater than or equal to 2k + 1. We determine k asymptotically for fixed k the minimum value n = e(k) for which this inequality holds. In addition we give an asymptotic formula for the solution of k Gamma(n) Gamma(n - 2k + 1) = Gamma(2)(n - k + 1) for n greater than or equal to 2k + 1, as k --> infinity, when n and k are not restricted to take integer values. We also show that for all prime powers q and n greater than or equal to 2k, k greater than or equal to 1, the q-analogues K-q(n, k) are Hamiltonian by consideration of the analogous inequality for q-binomial coefficients. (C) 1996 Academic Press, Inc.