Long paths and toughness of k-trees and chordal planar graphs

On the other hand, we present graphs whose longest paths are short. Namely, we construct 1-tough chordal planar graphs and 1-tough planar 3-trees, and we show that the shortness exponent of the class is 0, at most log3022, respectively. Both improve the bound of Böhme et al. Furthermore, the construction provides k-trees (for k≥4) of toughness greater than 1.