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) ( n&k k ) for n 2k+1. We determine 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 k1 (n) 1 (n&2k+1)= 1 (n&k+1) for n 2k+1, as k , when n and k are not restricted to take integer values. We also show that for all prime powers q and n 2k, k 1, the q-analogues Kq(n, k) are Hamiltonian by consideration of the analogous inequality for q-binomial coefficients. 1996 Academic Press, Inc.