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A graph G is minimally t -tough if the toughness of G is t and the deletion of any edge from G decreases the toughness. Kriesell conjectured that for every minimally 1 -tough graph the minimum degree δ ( G ) = 2 . We show that in every minimally 1 -tough graph δ ( G ) ≤ n 3 + 1 . We also prove that every minimally 1 -tough, claw-free graph is a cycle. On the other hand, we...
Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of 5 4 -tough maximal planar graphs presented by Harant and Owens (1995). In addition, we present two generalizations of a similar result of Tkáč who considered 1 -tough maximal planar graphs (Tkáč, 1996); we remark that one of these generalizations gives a tight...
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