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 54-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 upper bound. We fix a problematic argument used in both mentioned papers.