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Studying the shortness of longest cycles in maximal planar graphs, we improvethe upper bound on the shortness exponent of the class of $\frac{5}{4}$-toughmaximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995),301--305]. In addition, we present two generalizations of a similar result ofTk\'{a}\v{c} who considered $1$-tough maximal planar graphs [Discrete Math. 154(1996), 321--328]; we remark that one of these generalizations gives a tightupper bound. We fix a problematic argument used in the first paper.

An update on non-Hamiltonian $\frac{5}{4}$-tough maximal planar graphs