Papers › On a conjecture of Faudree and Schelp
On a conjecture of Faudree and Schelp
Jan Goedgebeur, Jorik Jooken, Michiel Provoost, Carol T. Zamfirescu
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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 ≥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.
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