Removable edges in a -connected graph and a construction method for -connected graphs

An edge e of a k -connected graph G is said to be a removable edge if G ⊖ e is still k -connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e and, for any end vertex of e with degree k − 1 in G − e , say x , deleting x and then adding edges between any pair of non-adjacent vertices in N G − e ( x ) . Xu and Guo [Liqiong Xu, Xiaofeng Guo, Removable edges in a 5-connected graph and a construction method of 5-connected graphs, Discrete Math. 308 (2008) 1726–1731] proved that a 5-connected graph G has no removable edge if and only if G ≅ K 6 , using this result, they gave a construction method for 5-connected graphs. A k -connected graph G is said to be a quasi ( k + 1 ) -connected if G has no nontrivial k -vertex cut. Jiang and Su [Hongxing Jiang, Jianji Su, Minimum degree of minimally quasi ( k + 1 ) -connected graphs, J. Math. Study 35 (2002) 187–193] conjectured that for k ≥ 4 the minimum degree of a minimally quasi k -connected graph is equal to k − 1 . In the present paper, we prove this conjecture and prove for k ≥ 3 that a k -connected graph G has no removable edge if and only if G is isomorphic to either K k + 1 or (when k is even) the graph obtained from K k + 2 by removing a 1-factor. Based on this result, a construction method for k -connected graphs is given.