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

TL;DR

This paper improves bounds on the shortness exponent of 5/4-tough maximal planar graphs, generalizes previous results, and corrects a prior argument, advancing understanding of cycle lengths in these graphs.

Contribution

It provides new upper bounds, generalizations of existing results, and fixes an error in earlier work on maximal planar graphs with specific toughness.

Findings

Improved upper bound on the shortness exponent for 5/4-tough maximal planar graphs

Two new generalizations of results on 1-tough maximal planar graphs

Correction of a problematic argument in previous literature

Abstract

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 [Discrete Math. 147 (1995), 301--305]. In addition, we present two generalizations of a similar result of Tk\'{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 tight upper bound. We fix a problematic argument used in the first paper.