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

TL;DR

This paper establishes conditions under which k-trees and chordal planar graphs are Hamilton-connected based on toughness, and constructs graphs with short longest paths, improving previous bounds and extending known results.

Contribution

It improves toughness-based Hamilton-connectedness results for k-trees and chordal planar graphs, and constructs graphs with short longest paths, extending bounds on shortness exponents.

Findings

k-trees with toughness > k/3 are Hamilton-connected for k ≥ 3

Chordal planar graphs with toughness > 1 are Hamilton-connected

Constructed graphs with short longest paths and improved bounds on shortness exponent

Abstract

We show that every $k$-tree of toughness greater than $\frac{k}{3}$ is Hamilton-connected for $k \geq 3$. (In particular, chordal planar graphs of toughness greater than $1$ are Hamilton-connected.) This improves the result of Broersma et al. (2007) and generalizes the result of B\"ohme et al. (1999). 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 $\log_{30}{22}$, respectively. Both improve the bound of B\"ohme et al. Furthermore, the construction provides $k$-trees (for $k \geq 4$) of toughness greater than $1$.