On an Inequality Arising in a Comparison of Hypercubes and Star Graphs

Gideon Ehrlich and Jeremy Schiff

We define an enkay pair to be a pair of positive integers (n,k) such that (n-1)(n-1)! < 2^k < n! . This inequality arises when comparing a hypercube with the smallest larger star graph. We list the 15 enkay pairs with n ≤ 10⁶, and speculate on whether the number of enkay pairs is finite or infinite.