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

- Received:

DOI: 10.1007/BF01305995

- Cite this article as:
- Fleischner, H. Monatshefte für Mathematik (1976) 82: 125. doi:10.1007/BF01305995

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## Abstract

The square*G*^{2} of a graph*G* has the same point set as*G*, and two points of*G*^{2} are adjacent in*G*^{2} if and only if their distance in*G* is at most two. The result that*G*^{2} is Hamiltonian if*G* is two-connected, has been established early in 1971. A conjecture (ofA. Bondy) followed immediately: Suppose*G*^{2} to have a Hamiltonian cycle; is it true that for any*v*∈*V(G)*, there exist cycles*C*_{j} containing*v* and having arbitrary length*j*, 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 if*G*^{2} contains a Hamiltonian path*P(v, w)* joining the points*v* and*w*, then*G*^{2} contains for any*j* with*d*_{G}^{2}*(v, w)*≤*j*≤≤|*V(G)*|−1 a path*P*_{j}*(v, w)* of length*j* joining*v* and*w*. 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) If*G* is two-connected, then*G*^{2} is Hamiltonian-connected; (b) If*G* is two-connected, then*G*^{2} is 1-Hamiltonian.