On the edge-connectivity of C_4-free graphs

Let $G$ be a connected graph of order $n$ and minimum degree $delta(G)$.The edge-connectivity $lambda(G)$ of $G$ is the minimum numberof edges whose removal renders $G$ disconnected. It is well-known that$lambda(G) leq delta(G)$,and if $lambda(G)=delta(G)$, then$G$ is said to be maximally edge-connected. A classical resultby Chartrand gives the sufficient condition $delta(G) geq frac{n-1}{2}$for a graph to be maximally edge-connected. We give lower bounds onthe edge-connectivity of graphs not containing $4$-cycles that implythat forgraphs not containing a $4$-cycle Chartrand's condition can be relaxedto $delta(G) geq sqrt{frac{n}{2}} +1$, and if the graphalso contains no $5$-cycle, or if it has girth at least six,then this condition can be relaxed further,by a factor of approximately $sqrt{2}$. We construct graphsto show that for an infinite number of values of $n$both sufficient conditions are best possible apart froma small additive constant.