On minimally 1-tough (K1 P4)-free graphs
Abstract
Agraph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases its toughness, where t is a positive real number. It is conjectured that every (K1 P4)-free 1-tough graph is hamiltonian. In this paper, we characterize the structure of minimally 1-tough (K1 P4)-free graphs, and thus show that the above conjecture is true for minimally 1-tough graphs. Furthermore, it is also proved that the Kriesell's conjecture which states that each minimally 1-tough graph has a vertex of degree 2 holds for minimally 1-tough (K1 P4)-free graphs.
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