Diameter, edge-connectivity, and C4-freeness

Abstract

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that d≤(2n−3)/5 for every connected C4-free graph of order n, diameter d, and edge-connectivity at least 3, which is best possible up to a small additive constant. For edge-connectivity at least 4, we improve this to d≤(n−1)/3. Furthermore, adapting a construction due to Erdős, Pach, Pollack, and Tuza, for an odd prime power q at least 7, and every positive integer k, we show the existence of a connected C4-free graph of order n=(q2+q−1)k+1, diameter d=4k, and edge-connectivity λ at least q−6, that is, d≥4(n−1)/(λ2+O(λ)) for the constructed graph.