Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G a e is still k-connected, where Ga 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 NG−e(x). Xu and Guo [28] proved that a 5-connected graph G has no removable edge if and only if G ∼= K6, 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 [11] 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 Kk+1 or (when k is even) the graph obtained from Kk+2 by removing a 1-factor. Based on this result, a construction method for k-connected graphs is given. ∗ The Project Supported by NSFC (No. 10831001), the 985 Invention Project on Information Technique of Xiamen University (2004-2007), the Natural Science Foundation of Guangxi (No. 0640063), and the Science-Technology Foundation for Young Scientists of Fujian (2007F3070). † Corresponding author. E-mail: [email protected]