On constructing snakes in powers of complete graphs

We prove the conjecture of Abbott and Katchalski that for every m ≥ 2 there is a positive constant λm such that S(K d mn) ≥ λmnS(K m ) where S(K m) is the length of the longest snake (cycle without chords) in the cartesian product K m of d copies of the complete graph Km. As a corollary, we conclude that for any finite set P of primes there is a constant c = c(P ) > 0 such that S(K n) ≥ cnd−1 for any n divisible by an element of P and any d ≥ 1. Supported by the WVU Senate Research Grant #R-93-033