On constructing snakes in powers of complete graphs Original Research Article

We prove the conjecture of Abbott and Katchalski that for every m ⩾ 2 there is a positive constant λm such that S(Kmnd) ⩾ λmnd − 1S(Kmd − 1) where S(Kmd) is the length of the longest snake (cycle without chords) in the cartesian product Kmd 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(Knd) ⩾ cnd − 1 for any n divisible by an element of P and any d ⩾ 1.