On hamiltonian cycles in the prism over the odd graphs

Bueno, Letícia Rodrigues; Horák, Peter

doi:10.1002/jgt.20550

Cited by 6 publications

(4 citation statements)

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“…Therefore, one can assert that a graph is prism-hamiltonian if it has a spanning good even cactus. This strategy is in fact the most common approach used to prove the prism-hamiltonicity of many planar and non-planar graph classes; we refer to [19,15,2,16,3,6,7,21] for examples. 1 It is worth noting that in [19,15,16,3,6] a more restrictive approach was adopted, namely to show the existence of a spanning good even cactus with maximum degree at most three, that is, any two cycles of the cactus are disjoint.…”

Section: Theorem 2 ([20]mentioning

confidence: 99%

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On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs

2022

Electron. J. Combin.

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The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. The graph $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become the most common technique in proving prism-hamiltonicity. Špacapan [S. Špacapan. A counterexample to prism-hamiltonicity of 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we answer his question in the negative, by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. In addition, we prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.

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Section: Theorem 2 ([20]mentioning

confidence: 99%

“…We denote by K i A the corresponding copy of K A in A i . It is straightforward to verify that 3 , sw 3 1 , tw 5 n+1 , sr 5 , tw 7 n+1 , tr 5 } is a spanning good even cactus of G.…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs

2022

Electron. J. Combin.

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“…Therefore, one can assert that a graph is prism-hamiltonian if it has a spanning good even cactus. This strategy has been used in proving prism-hamiltonicity for various planar and non-planar graph classes; we refer to [14,12,2,13,4,5,6,15] for examples. 1 It is worth noting that in [14,12,13,4,5] a more restrictive approach was adopted, namely showing the existence of a spanning good even cactus with maximum degree at most three.…”

Section: Theorem 2 ([16]mentioning

confidence: 99%

“…This strategy has been used in proving prism-hamiltonicity for various planar and non-planar graph classes; we refer to [14,12,2,13,4,5,6,15] for examples. 1 It is worth noting that in [14,12,13,4,5] a more restrictive approach was adopted, namely showing the existence of a spanning good even cactus with maximum degree at most three. This proof technique motivates us to refine the spanning structure hierarchy as follows:…”

Section: Theorem 2 ([16]mentioning

confidence: 99%

On the spanning structure hierarchy of 3-connected planar graphs

2021

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The prism over a graph G is the Cartesian product of G with the complete graph K 2 . G is prism-hamiltonian if the prism over G has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle, and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become one of the most common techniques in proving prism-hamiltonicity. Špacapan asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we give a negative answer to this question by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. We also prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.

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