On constructing snakes in powers of complete graphs (Q1381860)

The author proves a conjecture of H. L. Abbott and M. Katchalski that for every \(m\geq 2\) there is a positive constant \(\lambda_m\) such that \(S(K_{mn}^d)\geq \lambda_m n^{d-1} S(K_m^{d-1})\), where \(S(K_m^d)\) is the length of a longest chordless cycle (snake) in the Cartesian product \(K_m^d\) of \(d\) copies of the complete graph \(K_m\). As a corollary the author concludes that for any finite set \(P\) of primes there is a constant \(c= c(P)> 0\) such that \(S(K_n^d)\geq cn^{d-1}\) for any \(n\) divisible by an element of \(P\) and any \(d\geq 1\).