The Removable Edges and the Contractible Subgraphs of 5-Connected Graphs

An edge of a k-connected graph G is said to be k-removable if G − e is still k-connected. A subgraph H of a k-connected graph is said to be k-contractible if its contraction, that is, identification every component of H to a single vertex, results again a k-connected graph. In this paper, we show that there is either a removable edge or a contractible subgraph in a 5-connected graph which contains an edge with both endvertices have degree more than five. Thus every edge of minor minimal 5-connected graph is incident to at least one vertex of degree 5.