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Trees with Hamiltonian square

  • Frank Harary (a1) and Allen Schwenk (a2)
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Plummer (see [2; p. 69]) conjectured that the square of every block is hamiltonian, and this has just been proved by Fleischner [1]. It was shown by Karaganis [3] that the cube of every connected graph, and hence the cube of every tree, is hamiltonian. Our present object is to characterize those trees whose square is hamiltonian in three equivalent ways.

We follow the terminology and notation of the book [2]. In particular, the following concepts are used in stating our main result. A graph is hamiltonian if it has a cycle containing all its points. The graph with the same points as G, in which two points are adjacent if their distance in G is at most 2, is denoted by G2 and is called the square of G. The subdivision graph S(G) is formed (Figure 1) by inserting a point of degree two on each line of G.

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References
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1.Fleischner, H., “The square of every nonseparable graph is hamiltonian,” to appear.
2.Hararay, F., Graph Theory (Addison-Wesley, Reading, Mass. 1969).
3.Karaganis, J., “On the cube of a graph,” Canad. Math. Bull., 11 (1968), 295296.
4.Neumann, F., “On a certain ordering of the vertices of a tree,” Casopis Pest. Mat., 89 (1964), 323339.
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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