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

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that $d \leq (2n-3)/5$ 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 $d \leq (n-3)/3$. Furthermore, adapting a construction due to Erd\H{o}s, 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 $\lambda$ at least $q-6$, in particular, $d\geq 4(n-1)/(\lambda^2+O(\lambda))$.