An update on non-Hamiltonian 54-tough maximal planar graphs

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 and Owens (1995). In addition, we present two generalizations of a similar result of Tká? who considered $1$-tough maximal planar graphs (Tká?, 1996); we remark that one of these generalizations gives a tight upper bound. We fix a problematic argument used in both mentioned papers.