Codiameters of 3-domination critical graphs with toughness more than one

A graph G is 3-domination-critical (3-critical, for short), if its domination number γ is 3 and the addition of any edge decreases γ by 1. In this paper, we show that every 3-critical graph with independence number 4 and minimum degree 3 is Hamilton-connected. Combining the result with those in [Y.J. Chen, F. Tian, B. Wei, Hamilton-connectivity of 3-domination critical graphs with α ≤ δ , Discrete Mathematics 271 (2003) 1–12; Y.J. Chen, F. Tian, Y.Q. Zhang, Hamilton-connectivity of 3-domination critical graphs with α = δ + 2 , European Journal of Combinatorics 23 (2002) 777–784; Y.J. Chen, T.C.E. Cheng, C.T. Ng, Hamilton-connectivity of 3-domination critical graphs with α = δ + 1 ≥ 5 , Discrete Mathematics 308 (2008) (in press)], we solve the following conjecture: a connected 3-critical graph G is Hamilton-connected if and only if τ ( G ) > 1 , where τ ( G ) is the toughness of G .