An Infinite Family of Planar Hypohamiltonian Oriented Graphs

Aardt, Susan A. van; Burger, Alewyn P.; Frick, Marietjie

doi:10.1007/s00373-012-1165-z

Cited by 8 publications

(12 citation statements)

References 3 publications

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“…≠ K 1,3 ) 3-fragments of planar hypohamiltonian graphs, which we will call hypo-fragment in the following. For a characterisation of such fragments, we need to add to (1) and (2) , so it has girth 3. Every vertex of a triangle in B has degree at least 4: for the vertices in X , this follows from Lemma 4, and for all other triangles this follows from Lemma 3.…”

Section: F I G U R E 2 a Plane Hypohamiltonian Graph Containing 30 Cumentioning

confidence: 99%

“…For results treating the planar case and which are not included in [15], we refer to the works of Aldred, Bau, Holton, and McKay [3], the author and Zamfirescu [35,36], Araya and Wiener [4,33], Jooyandeh, McKay, Östergård, Pettersson, and the author [16], McKay [19], Goedgebeur and the author [11,12], and Wiener [32]. For the situation in directed graphs, see for instance van Aardt et al [1], which answers affirmatively Thomassen's [26,Question 9] from 1976 whether planar hypohamiltonian oriented graphs exist.…”

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confidence: 99%

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Cubic vertices in planar hypohamiltonian graphs

Zamfirescu

2018

Journal of Graph Theory

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Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex‐deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.

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Section: F I G U R E 2 a Plane Hypohamiltonian Graph Containing 30 Cumentioning

confidence: 99%

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confidence: 99%

Cubic vertices in planar hypohamiltonian graphs

Zamfirescu

2018

Journal of Graph Theory

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“…We conjecture that, in every oriented graph with circumference k ≤ 7, one can destroy all the longest cycles by removing 1/k of the vertices. (We know that the smallest hypohamiltonian oriented graph has circumference 8 -see [1].) 00 00 11 11 00 11 00 11 000 111 000 000 111 111 000 111 Figure 4: The case r = 1.…”

Section: Definition 33 B ∈ B If and Only If Each Of The Following Hmentioning

confidence: 99%

Destroying longest cycles in graphs and digraphs

Aardt

Burger

Dunbar

et al. 2015

Discrete Applied Mathematics

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In 1978, C. Thomassen [Hypohamiltonian graphs and digraphs, Proceedings of the International Conference on the Theory and Applications of Graphs, Kalamazoo, 1976, Springer Verlag, pp. 557 -571] proved that in any graph one can destroy all the longest cycles by deleting at most one third of the vertices. We show that for graphs with circumference k ≤ 8 it suffices to remove at most 1/k of the vertices. The Petersen graph demonstrates that this result cannot be extended to include k = 9 but we show that in every graph with circumference nine we can destroy all 9-cycles by removing 1/5 * Corresponding author. of the vertices. We consider the analogous problem for digraphs and show that for digraphs with circumference k = 2, 3, it suffices to remove 1/k of the vertices. However this does not hold for k ≥ 4.

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“…Using a result from [105], Thomassen [130] disproves the old conjecture of Adám [3] that any digraph containing a directed cycle has an arc whose reversal decreases the total number of directed cycles. For articles published recently-these also provide overviews-see [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…For results therein, see for instance Thomassen's paper [127] and 1 Let k ≥ 1. A graph G is k-hamiltonian if for any S ⊂ V (G) with |S| ≤ k we have that G − S is hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

On Hypohamiltonian and Almost Hypohamiltonian Graphs

Zamfirescu

2014

Journal of Graph Theory

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A graph G is almost hypohamiltonian if G is non‐hamiltonian, there exists a vertex w such that G−w is non‐hamiltonian, and for any vertex v≠w the graph G−v is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4‐connected almost hypohamiltonian graph, while Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every n≥74. During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.

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An Infinite Family of Planar Hypohamiltonian Oriented Graphs

Cited by 8 publications

References 3 publications

Cubic vertices in planar hypohamiltonian graphs

Cubic vertices in planar hypohamiltonian graphs

Destroying longest cycles in graphs and digraphs

On Hypohamiltonian and Almost Hypohamiltonian Graphs

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