Seven Problems on Hypohamiltonian and Almost Hypohamiltonian Graphs

Zamfirescu, Carol T.

doi:10.1007/978-3-319-28186-5_21

Cited by 5 publications

(12 citation statements)

References 3 publications

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“…This answers [31,Problem 6(a)] and [31,Problem 7]. We also prove that such a graph of order n exists for every n ≥ 73, partially answering [31,Problem 6(b)]. Furthermore, we show that the smallest planar almost hypohamiltonian graph of girth 5 has order 44.…”

Section: Introductionsupporting

confidence: 58%

“…This article is structured as follows. In Section 2 we extend a series of results (and settle certain open questions) of the second author [30,31]. In particular, in Section 2.1 we present a specialised algorithm for generating all almost hypohamiltonian graphs.…”

Section: Introductionmentioning

confidence: 69%

“…In Section 2.4 we give a lower bound for the order of the smallest planar almost hypohamiltonian graphs, and improve the upper bound for the order of the smallest planar almost hypohamiltonian graph whose exceptional vertex is cubic (from 47, see [30], to 36), which are particularly important due to results from [30]. This answers [31,Problem 6(a)] and [31,Problem 7]. We also prove that such a graph of order n exists for every n ≥ 73, partially answering [31,Problem 6(b)].…”

Section: Introductionmentioning

confidence: 75%

“…Until recently, the smallest known planar almost hypohamiltonian graph had order 39, see [30]. [30,Problem 3] and [31,Problem 6] ask for the smallest order of a planar almost hypohamiltonian graph, and the smallest number n 0 such that there exists a planar almost hypohamiltonian graph of order n for every n ≥ n 0 . Although we are not able to fully settle these questions, we can report progress on both the lower and upper bounds of the order of the smallest planar almost hypohamiltonian graph, as well as an improvement of n 0 .…”

Section: The Planar Casementioning

confidence: 99%

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On almost hypohamiltonian graphs

Goedgebeur,

Zamfirescu

2016

Preprint

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A graph G is almost hypohamiltonian (a.h.) if G is non-hamiltonian, there exists a vertex w in G such that G − w is non-hamiltonian, and G − v is hamiltonian for every vertex v = w in G. The second author asked in [J. Graph Theory 79 (2015) 63-81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, Östergård, Pettersson, and the second author.

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Section: Introductionsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 75%

Section: The Planar Casementioning

confidence: 99%

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On almost hypohamiltonian graphs

Goedgebeur,

Zamfirescu

2016

Preprint

Self Cite

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“…To generate networks with desired topology and attributes, the existing SNSs simulate the unobservable (represented with a ) and partially observable network components (represented with a ) based on the observable network components (represented with a ) and an inner rule that directs the network growth (see Table 1). [6], [20], [80], [24], [2], [3], [71], [73], [69], [22], [54], [9], [10], [83], [59], [33], [1], [39], [32], [57], [11], [30], [23], [31] Hybrid [53], [36], [5], [74], [4], [7], [8], [72], [55] [49], [52], [79], [50] [75], [34], [29] [28], [76] [15], [18] As is shown in Table 1, the simulated networks are built with purely simulated topology and attributes, dealing with the unobservability problem of all network components for the analysis of the topological features [1,6,…”

Section: Current State-of-the-art Of Social Network Simulatorsmentioning

confidence: 99%

Toward Digital Twin Oriented Modeling of Complex Networked Systems and Their Dynamics: A Comprehensive Survey

2022

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This paper aims to provide a comprehensive critical overview on how entities and their interactions in Complex Networked Systems (CNS) are modelled across disciplines as they approach their ultimate goal of creating a Digital Twin (DT) that perfectly matches the reality. We propose four complexity dimensions for the network representation and five generations of models for the dynamics modelling to describe the increasing complexity level of the CNS that will be developed towards achieving DT (e.g. CNS dynamics modelled offline in the 1st generation v.s. CNS dynamics modelled simultaneously with a twoway real time feedback between reality and the CNS in the 5th generation). Based on that, we propose a new framework to conceptually compare diverse existing modelling paradigms from different perspectives and create unified assessment criteria to evaluate their respective capabilities of reaching such an ultimate goal. Using the proposed criteria, we also appraise how far the reviewed current state-of-the-art approaches are from the idealised DTs. Finally, we identify and propose potential directions and ways of building a DT-orientated CNS based on the convergence and integration of CNS and DT utilising a variety of crossdisciplinary techniques.

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New constructions of hypohamiltonian and hypotraceable graphs

Wiener

2017

Journal of Graph Theory

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A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable, but all vertex‐deleted subgraphs of G are hamiltonian/traceable. All known hypotraceable graphs are constructed using hypohamiltonian graphs; here we present a construction that uses so‐called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex‐deleted subgraphs are hamiltonian with exactly one exception, see [15]). This construction is an extension of a method of Thomassen [11]. As an application, we construct a planar hypotraceable graph of order 138, improving the best‐known bound of 154 [8]. We also prove a structural type theorem showing that hypotraceable graphs possessing some connectivity properties are all built using either Thomassen's or our method. We also prove that if G is a Grinbergian graph without a triangular region, then G is not maximal nonhamiltonian and using the proof method we construct a hypohamiltonian graph of order 36 with crossing number 1, improving the best‐known bound of 46 [14].

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Seven Problems on Hypohamiltonian and Almost Hypohamiltonian Graphs

Cited by 5 publications

References 3 publications

On almost hypohamiltonian graphs

On almost hypohamiltonian graphs

Toward Digital Twin Oriented Modeling of Complex Networked Systems and Their Dynamics: A Comprehensive Survey

New constructions of hypohamiltonian and hypotraceable graphs

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