In the square of graphs, hamiltonicity and pancyclicity, Hamiltonian connectedness and panconnectedness are equivalent concepts

Abstract

The squareG ² of a graphG has the same point set asG, and two points ofG ² are adjacent inG ² if and only if their distance inG is at most two. The result thatG ² is Hamiltonian ifG is two-connected, has been established early in 1971. A conjecture (ofA. Bondy) followed immediately: SupposeG ² to have a Hamiltonian cycle; is it true that for anyv∈V(G), there exist cyclesC _j containingv and having arbitrary lengthj, 3≤j≤|V(G)|. The proof of this conjecture is one of the two main results of this paper. The other main result states that ifG ² contains a Hamiltonian pathP(v, w) joining the pointsv andw, thenG ² contains for anyj withd _G ² (v, w)≤j≤≤|V(G)|−1 a pathP _j (v, w) of lengthj joiningv andw. By this, a conjecture ofF. J. Faudree andR. H. Schelp is proved and generalized for the square of graphs.

However, to prove these two results extensive preliminary work is necessary in order to make the proof of the main results transparent (Theorem 1 through 5); and Theorem 3 plays a central role for the main results. As can be seen from the statement of Theorem 3, the following known results follow in a stronger form: (a) IfG is two-connected, thenG ² is Hamiltonian-connected; (b) IfG is two-connected, thenG ² is 1-Hamiltonian.