It is proved that for any " > 0 there is a 1-tough chordal planar graph G " such that the length of a longest cycles of G " is less than "jV (G)j".Expand

Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough maximal planar graphs presented by Harant… Expand

The relation between Hamiltonicity and toughness of a graph is a long standing research problem and it is shown that the following three statements are equivalent: (i) P_n\square H is Hamiltonian; (ii) H is-tough; and (iii) $H$ has a path factor.Expand

This survey has attempted to bring together most of the results and papers that deal with toughness related to cycle structure into a few self explanatory categories.Expand

In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology… Expand