Systematic searches for hypohamiltonian graphs

Collier, James B.; Schmeichel, E. F.

doi:10.1002/net.3230080303

Cited by 18 publications

(7 citation statements)

References 7 publications

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“…In Theorem 5, we present a stronger result. As a further corollary we obtain a strengthening of (11, Proposition 2.6 [ii]), for which we need the following lemma (which strengthens an observation of Collier and Schmeichel [9]). In [25], no proof is givenwe have seen a proof in the penultimate paragraph of the proof of Lemma 2.…”

Section: Planar Hypohamiltonian Graphssupporting

confidence: 53%

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: Planar Hypohamiltonian Graphssupporting

confidence: 53%

Cubic vertices in planar hypohamiltonian graphs

Zamfirescu

2018

Journal of Graph Theory

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“…However, for every k ≥ 23 there exists a hypohamiltonian graph containing the complete bipartite graph K 2k−44,2k−44 , as proven by Thomassen [39]. [13]). Let G be a hypohamiltonian graph containing a triangle T .…”

Section: Preparationmentioning

confidence: 89%

Improved bounds for hypohamiltonian graphs

Goedgebeur

Zamfirescu

2017

Ars Math. Contemp.

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A graph G is hypohamiltonian if G is non-hamiltonian and G − v is hamiltonian for every v ∈ V (G). In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (iii) the smallest cubic planar graph has order at least 54, and (iv) the smallest cubic planar graph of girth 5 with non-trivial automorphism group has order 78. Important note: the version of this manuscript which was published in [15] contained an error in the definition of good A-edge on page 239. We fixed this error in the current manuscript and also adjusted the code of our generator for hypohamiltonian graphs accordingly [16]. Because of this error, theoretically some hypohamiltonian graphs might have been missed by the program. However, we reran all computations which were reported in [15] and no graphs were missed by the old version of the program.

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“…Let e * (1,10) and note in Fig.1 that V(PG-e) has three similarity classes, namely {l,10}, {2,4,7,9} and {3,5,6,8}. Henoe PG-e is HT since it has the Haailtonian paths [1,4,5,6,7,8,9,10,2,3] and [1,4,3.8,7,6,5,9,10,2].…”

Section: Definitionmentioning

confidence: 99%