Diameter, edge-connectivity, and $C_4$-freeness

Abstract

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that d ≤ (2n−3)/5 for every connected C 4 -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 − 3)/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 C 4 -free graph of order n = (q 2 + q − 1)k + 1, diameter d = 4k, and edge-connectivity λ at least q − 6, in particular, d ≥ 4(n − 1)/(λ 2 + O(λ)).