In the square of graphs, hamiltonicity and pancyclicity, Hamiltonian connectedness and panconnectedness are equivalent concepts
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The squareG2 of a graphG has the same point set asG, and two points ofG2 are adjacent inG2 if and only if their distance inG is at most two. The result thatG2 is Hamiltonian ifG is two-connected, has been established early in 1971. A conjecture (ofA. Bondy) followed immediately: SupposeG2 to have a Hamiltonian cycle; is it true that for anyv∈V(G), there exist cyclesCj 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 ifG2 contains a Hamiltonian pathP(v, w) joining the pointsv andw, thenG2 contains for anyj withdG2(v, w)≤j≤≤|V(G)|−1 a pathPj(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, thenG2 is Hamiltonian-connected; (b) IfG is two-connected, thenG2 is 1-Hamiltonian.
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