Contractible edges in triangle-free graphs (Q1085181)

An edge \(e=xy\) of finite simple graph G is called k-contractible if the contraction of the edge results in a k-connected graph. In this paper the authors nicely extend previous results of Thomassen that every k- connected graph of girth at least four has a k-contractible edge, and of Thomassen and Toft that every 3-connected graph of girth at least four has a cycle consisting of 3-contractible edges, to show that if \(k\geq 2\) then every k-connected graph of girth at least four contains at least \(| G| +3k^ 2/2-3k\) k-contractible edges.