%0 Journal Article %T On the edge-connectivity of C_4-free graphs %J Communications in Combinatorics and Optimization %I Azarbaijan Shahid Madani University %Z 2538-2128 %A Dankelmann, Peter %D 2019 %\ 12/01/2019 %V 4 %N 2 %P 141-150 %! On the edge-connectivity of C_4-free graphs %K edge-connectivity %K maximally edge-connected %K graph %R 10.22049/cco.2019.26453.1113 %X Let $G$ be a connected graph of order $n$ and minimum degree $\delta(G)$. The edge-connectivity $\lambda(G)$ of $G$ is the minimum number of 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 result by Chartrand gives the sufficient condition $\delta(G) \geq \frac{n-1}{2}$ for a graph to be maximally edge-connected. We give lower bounds on the edge-connectivity of graphs not containing $4$-cycles that imply that for graphs not containing a $4$-cycle Chartrand's condition can be relaxed to $\delta(G) \geq \sqrt{\frac{n}{2}} +1$, and if the graph also 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 graphs to show that for an infinite number of values of $n$ both sufficient conditions are best possible apart from a small additive constant. %U http://comb-opt.azaruniv.ac.ir/article_13856_bcb7a6a4d152a475da2d7bdfa37a581e.pdf