Infinite families of 2-hypohamiltonian/2-hypotraceable oriented graphs

Aardt, Susan A. van; Burger, Alewyn P.; Frick, Marietjie; Llano, Bernardo; Zuazua, Rita

doi:10.1007/s00373-013-1312-1

Cited by 10 publications

(9 citation statements)

References 15 publications

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“…Proof. We shall prove (1), (3) and (5). The proofs of (2) and ( 4) are similar to those of (1) and (3), respectively.…”

Section: Lemmasupporting

confidence: 59%

“…However, for k = 7 and for every k 9 there exist k-traceable oriented graphs of order k + 1 that are nontraceable, as shown in [6]. There also exist nontraceable k-traceable oriented graphs of order k + 2 for infinitely many k, as shown in [5], but the following theorem shows that the order of nontraceable k-traceable oriented graphs is bounded above by a function of k. Theorem 3. [2,4] Let k 7 and suppose D is a k-traceable oriented graph of order n and independence number α.…”

mentioning

confidence: 99%

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An Iterative Approach to the Traceability Conjecture for Oriented Graphs

Aardt

Burger

Dunbar

et al. 2013

Electron. J. Combin.

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A digraph is $k$-traceable if its order is at least $k$ and each of its subdigraphs of order $k$ is traceable. The Traceability Conjecture (TC) states that for $k\geq 2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. It has been shown that for $2\leq k\leq 6$, every $k$-traceable oriented graph is traceable. We develop an iterative procedure to extend previous results regarding the TC. In particular, we prove that every $7$-traceable oriented graph of order at least 9 is traceable and every 8-traceable graph of order at least 14 is traceable.

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“…Proof. We shall prove (1), (3) and (5). The proofs of (2) and ( 4) are similar to those of (1) and (3), respectively.…”

Section: Lemmasupporting

confidence: 59%

mentioning

confidence: 99%

An Iterative Approach to the Traceability Conjecture for Oriented Graphs

Aardt

Burger

Dunbar

et al. 2013

Electron. J. Combin.

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“…We say that

G

∣ W ∣

‐ hypohamiltonian . (Note that van Aardt et al define in an

r

‐hypohamiltonian di graph differently.) A vertex from

W

is called exceptional .…”

Section: Introductionmentioning

confidence: 99%

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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“…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%

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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Infinite families of 2-hypohamiltonian/2-hypotraceable oriented graphs

Cited by 10 publications

References 15 publications

An Iterative Approach to the Traceability Conjecture for Oriented Graphs

An Iterative Approach to the Traceability Conjecture for Oriented Graphs

Cubic vertices in planar hypohamiltonian graphs

On Hypohamiltonian and Almost Hypohamiltonian Graphs

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