Revisiting the Hamiltonian theme in the square of a block: the general case

Abstract

This is the second part of joint research in which we show that every 2-connected graph G has the F 4 property. That is, given distinct x i

A best possible result for the square of a 2-block to be hamiltonian

“…We introduced the concept of the H k property and proved that every 2-block has the H 4 property but not the H 5 property in general. Similarly in [8] it is proved that every 2-block has the F 4 property but not the F 5 property in general.…”

“…Also very recently it was shown in [3] and [8] that a 2-block has the F 4 property; that is, given vertices x 1 , x 2 , x 3 , x 4 in the 2-block G, there is a hamiltonian path in G 2 joining x 1 and x 2 and traversing distinct edges x 3 y 3 and x 4 y 4 of G (see Theorem 7). The proof of this result is very long and is based on techniques developed by Fleischner in [5][6][7] and by Fleischner and Hobbs in [9].…”

The most general structure of graphs with hamiltonian or hamiltonian connected square

“…More recently, an algorithm yielding a hamiltonian cycle in the square of a 2-block in linear time, was developed, [1]. The methods developed in these much shorter proofs (including the algorithm just mentioned) do not seem to yield short proofs of Theorems 4 and 5 below, [4], [8]. These latter theorems are, on the other hand, instrumental in proving the central results of this paper, i.e., Theorems 1 and 2, and related algorithms.…”

“…The hamiltonian theme in the square of a 2-block has been recently revisited ( [3], [4], [8]), yielding the following results which are essential for this paper.…”

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On the basis of recent results on hamiltonicity, [4], and hamiltonian connectedness, [8], in the square of a 2-block, we determine the most general block-cutvertex structure a graph G may have in order to guarantee that G 2 is hamiltonian, hamiltonian connected, respectively. Such an approach was already developed in [9] for hamiltonian total graphs.

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