# On the edge-connectivity of C_4-free graphs

Document Type : Original paper

Author

University of Johannesburg

Abstract

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.

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