Revisiting the Hamiltonian theme in the square of a block: the case of $DT$-graphs

Chia, G. L.; Ekstein, Jan; Fleischner, Herbert

doi:10.4310/joc.2018.v9.n1.a7

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We study the squares of S(K 1,k+1 )-free graphs and their 2-connected spanning subgraphs of maximum degree at most k. We view the results of Harary and Schwenk (1971) and Henry and Vogler (1985) as the case k = 2 of this study, and we generalize these results by considering greater k.In this note, we continue the long-established and thorough study of Hamiltonian properties of the squares of graphs (for instance, see [9,2]). We recall that the square of a graph G is the graph on the same vertex set as G in which two vertices are adjacent if and only if their distance in G is either 1 or 2, and we let G 2 denote the this graph.

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mentioning

confidence: 84%

Trestles in the squares of graphs

Kabela,

Teska

2020

Preprint

0

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We study the squares of S(K 1,k+1 )-free graphs and their 2-connected spanning subgraphs of maximum degree at most k. We view the results of Harary and Schwenk (1971) and Henry and Vogler (1985) as the case k = 2 of this study, and we generalize these results by considering greater k.In this note, we continue the long-established and thorough study of Hamiltonian properties of the squares of graphs (for instance, see [9,2]).We recall that the square of a graph G is the graph on the same vertex set as G in which two vertices are adjacent if and only if their distance in G is either 1 or 2, and we let G 2 denote the this graph. We recall that a k-trestle (sometimes called k-covering) is a 2-connected spanning subgraph of maximum degree at most k. Clearly, 2-trestles are Hamilton cycles; and k-trestles are viewed as an extension of Hamiltonicity (for instance, see [8] and the references therein).We let S(K 1,k ) denote the graph obtained from K 1,k by subdividing each of its edges once (see Figure 1). Clearly, the square of S(K 1,k+1 ) has no ktrestle. We recall that Neuman [9] (and also Harary and Schwenk [6]) showed that for trees, being S(K 1,3 )-free (and having at least 3 vertices) is a necessary and sufficient condition in relation to Hamiltonicity of the square. Later, Henry and Vogler [7] showed that this condition is sufficient for all graphs (and this result was strengthened by Abderrezzak, Flandrin and Ryjáček [1] who studied additional properties of induced copies of S(K 1,3 )).We study the squares of S(K 1,k+1 )-free graphs and their k-trestles. For k = 3, we show the following.

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“…In this note, we continue the long-established and thorough study of Hamiltonian properties of the squares of graphs (for instance, see [9,2]).…”

mentioning

confidence: 84%

Trestles in the squares of graphs

Kabela

¹

,

Teska

²

2021

Discrete Mathematics

0

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“…The main result of [3] is the following result which is the larger part of the proof of Theorem 4 below.…”

Section: Introductionmentioning

confidence: 99%

“…This is the second part of joint research in which we establish the most general result for the square of a block (i.e., a 2-connected graph) to be hamiltonian connected. In the first part this was achieved in [3,Theorem 2] for the case of DT-graphs (i.e., graphs in which every edge is incident to a vertex of degree two). In the past, the approach to deal with 2-connected DT-graphs first and then generalize the corresponding results to blocks in general, was a logical consequence of the proof methods developed in [6]- [9], say.…”

Section: Introductionmentioning

confidence: 99%