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

Let ³(G) be the domination number of graph G, thus a graph G is k-edge-critical if ³ (G) = k, and for every nonadjacent pair of vertices u and υ, ³(G + uυ) = k-1. In Chapter 16 of the book "Domination in GraphsAdvanced 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 = 4 by constructing a class of 3-connected 4-edge-critical non-Hamiltonian graphs. © 2005 Wiley Periodicals, Inc.