Some 3-connected 4-edge-critical non-Hamiltonian graphs

Let (G) be the domination number of graph G, thus a graph G is k -edge-critical if (G) 1⁄4 k ; and for every nonadjacent pair of vertices u and v, (Gþ uv) 1⁄4 k 1. In Chapter 16 of the book ‘‘Domination in Graphs— Advanced Topics,’’ D. Sumner cites a conjecture of E. Wojcicka under the form ‘‘3-connected 4-critical graphs are Hamiltonian and perhaps, in general (i.e., for any k 4), (k 1)-connected, k -edge-critical graphs are Hamiltonian.’’ In this paper, we prove that the conjecture is not true for k 1⁄4 4 by constructing a class of 3-connected 4-edge-critical non-Hamiltonian graphs. 2005 Wiley Periodicals, Inc. J Graph Theory 50: 316–320, 2005