On a conjecture of Faudree and Schelp

TL;DR

This paper explores the limitations of hamiltonian-connected graphs by constructing specific counterexamples, including cubic planar graphs and graphs with gaps in their cycle spectra, challenging previous conjectures.

Contribution

It provides new cubic planar counterexamples and an infinite family of graphs with gaps in their cycle spectra, advancing understanding of hamiltonian-connected graph properties.

Findings

Existence of cubic planar counterexamples with no long odd paths

Construction of graphs with gaps in their cycle spectra

Counterexamples challenge previous conjectures about path lengths

Abstract

In 1976 Faudree and Schelp conjectured that in a hamiltonian-connected graph on $n$ vertices, any two distinct vertices are connected by a path of length $k$ for every $k \ge n/2$. In 1978 Thomassen constructed a (non-cubic and non-planar) family of counterexamples, showing that there exist hamiltonian-connected $n$-vertex graphs containing two vertices with no path of length $n-2$ between them. We complement this result by describing cubic planar counterexamples on $6p+16$ vertices, each containing vertices between which there is no path of any odd length greater than $1$ and at most $4p+9$. Motivated by a remark of Thomassen about a gap in the cycle spectrum of hamiltonian-connected graphs, we also describe an infinite family of hamiltonian-connected graphs with many gaps in the first half of their cycle spectra.