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Detour-saturated graphs of small girths
[article]

2018
*
arXiv
*
pre-print

(2) Let Pr be the

arXiv:1806.06564v1
fatcat:e7gorwdxojg3fi3b44t3lwdmx4
*graph*obtained from the Petersen*graph*by splitting*one*of its vertices into three leaves. Is Pr the*smallest*triangle-free detour-saturated*graph*? ... Frick asked the following three questions in 2005. (1) What is the*smallest*order of a detour-saturated*graph*of girth 4? ... 3, Proposition 6] proved that CT is*maximally**nonhamiltonian*. ...##
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A catalogue of small maximal nonhamiltonian graphs

1982
*
Discrete Mathematics
*

Special attention is paid to

doi:10.1016/0012-365x(82)90145-5
fatcat:ryi57476wzfbfiuzgujpjnolna
*maximal**nonhamiltonian**graphs*with non-positive scattering number since all remaining*ones*(with scatterirg number 1) are fully characterized and counted by the third auth ... Revised 15 Augusr 1979 and 5 January 1981 In this paper a catalogue of all*maximal**nonhamiltonian**graphs*of orders up to 10 is provided. ...##
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Maximum nonhamiltonian tough graphs

1991
*
Discrete Mathematics
*

Skupieri, Maximum

doi:10.1016/0012-365x(91)90314-r
fatcat:gv4eyany2jc2zbdgr5r5nardpe
*nonhamiltonian*tough*graphs*, Discrete Mathematics % (1991) 213-220. ... Notice that M,, found by Chvatal, is the*smallest*nontrivially*nonhamiltonian**graph*among tough*graphs*. ... Let G,=K,*K,*(3K,*-+K,), which is a*maximally**nonhamiltonian*tough*graph*of order n = 9 exhibited in [4] . ...##
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Subgraphs intersecting any hamiltonian cycle

1988
*
Journal of combinatorial theory. Series B (Print)
*

In terms of the

doi:10.1016/0095-8956(88)90097-4
fatcat:sk3mmjeei5ch7epjn62hkjqgfq
*maximal**nonhamiltonian**graphs*Corollary 1 translates into the following. COROLLARY 2. ... Then there exists a*graph*G*on*n vertices which is hypohamiltonian,*maximal**nonhamiltonian*in which for any nonaGacent vertices x, y there exist two hamiltonian x -y paths x.. uy, x.. vy, u # v, triangle ...##
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Toughness and nonhamiltonicity of polyhedral graphs

1993
*
Discrete Mathematics
*

Tutte the toughness t(G) of a

doi:10.1016/0012-365x(93)90519-y
fatcat:kwk7fodycrdtrbgv3gjfja5w2q
*nonhamiltonian*polyhedral*graph*G is less than or equal to $. ... A nonhamiitonian planar*graph*G*on*at least 5 vertices contains a set S of at most three vertices such that w(G -S) 2 2. Thus, we have t(G) c 2. ... (i) There is a*nonhamiltonian*Sregular polyhedral*graph**on*1254 vertices with toughness 2. ...##
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How tough is toughness?

2015
*
Bulletin of the European Association for Theoretical Computer Science
*

We survey results and open problems related to the toughness of

dblp:journals/eatcs/Broersma15
fatcat:di3kcho5efbktnh73wp4b2ac7e
*graphs*. ... Dillencourt [39] has also inquired about the complexity of recognizing 1tough,*maximal*planar*graphs*, noting that recognizing hamiltonian,*maximal*planar*graphs*is NP-hard. ...*nonhamiltonian**graphs*with an arbitrarily high toughness exist. ...##
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The cost of perfection for matchings in graphs

2016
*
Discrete Applied Mathematics
*

cubic

doi:10.1016/j.dam.2014.12.006
fatcat:jmzajjb4r5bnhawz6soavkbgty
*graphs*. ... Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly*on*bridgeless ... We show that the*smallest*bipartite planar*nonhamiltonian*bridgeless cubic*graph*[4] has η ≤ 1/2. Therefore, the class of*nonhamiltonian*bipartite cubic*graphs*has 1/3 ≤ η ≤ 1/2. ...##
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The maximum size of a graph with prescribed order, circumference and minimum degree
[article]

2021
*
arXiv
*
pre-print

All these results

arXiv:2107.01416v3
fatcat:23rhy432xja2tgrjj5gepwwxoq
*on*the size of*graphs*have clique versions. ... Erdős determined the maximum size of a*nonhamiltonian**graph*of order n and minimum degree at least k in 1962. Recently, Ning and Peng generalized. ... The author is grateful to Professor Xingzhi Zhan for suggesting the problems and for helpful discussions, and to*one*referee for her (his) kind suggestions. ...##
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Maximal Nontraceable Graphs with Toughness less than One

2008
*
Electronic Journal of Combinatorics
*

A

doi:10.37236/742
fatcat:k67nq7rv4fca5lc2zgvbr6fd4y
*graph*$G$ is*maximal*nontraceable (MNT) if $G$ does not have a hamiltonian path but, for every $e\in E\left( \overline{G}\right) $, the*graph*$G+e$ has a hamiltonian path. ... A*graph*$G$ is 1-tough if for every vertex cut $S$ of $G$ the number of components of $G-S$ is at most $|S|$. We investigate the structure of MNT*graphs*that are not 1-tough. ... Acknowledgement This paper contains results from the thesis of the third author which resulted from work done*on*toughness of MNT*graphs*at the Detour Workshops held at Salt Rock, South Africa in 2004 ...##
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Strongly pancyclic and dual-pancyclic graphs

2009
*
Discussiones Mathematicae Graph Theory
*

Say that a cycle C almost contains a cycle C − if every edge except

doi:10.7151/dmgt.1429
fatcat:ei6hg6lsdfbcpciixymtr26lhq
*one*of C − is an edge of C. ... This is equivalent to requiring, in addition, that the sizes of C − and C + differ by*one*from the size of C. ... A 2-connected*graph*is strongly pancyclic if and only if it satisfies both of the following: (1.1) Every nontriangular cycle almost contains a cycle of length*one*less. (1.2) Every*nonhamiltonian*cycle ...##
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Planar cubic hypohamiltonian and hypotraceable graphs

1981
*
Journal of combinatorial theory. Series B (Print)
*

Infinite families of planar cubic hypohamiltonian and hypotraceable

doi:10.1016/0095-8956(81)90089-7
fatcat:m6maby5libggppn3ileszitv5a
*graphs*are described and these are used to prove that the maximum degree and the maximum number of edges in a hypohamiltonian*graph*with ... Also, the possible order of a cubic hypohamiltonian*graph*is determined. L' (i -2)(d; -d:') = 0. ... It is easy to see that no vertex of a hypohamiltonian*graph*of order n has degree >f(n -1) and hence the*graph*has less than in' edges. ...##
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On properties of maximal 1-planar graphs

2012
*
Discussiones Mathematicae Graph Theory
*

A

doi:10.7151/dmgt.1639
fatcat:fefldaaa6nhthkowkdpz5f4k7m
*graph*is called 1-planar if there exists a drawing in the plane so that each edge contains at most*one*crossing. ... We study*maximal*1-planar*graphs*from the point of view of properties of their diagrams, local structure and hamiltonicity. ... The following construction shows that there are*nonhamiltonian*4-connected 1-planar*graphs*: take the plane drawing of the Barnette-Bosák-Lederberg*graph*(that is, the*smallest**nonhamiltonian*cubic planar ...##
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Non-hamiltonian triangulations with distant separating triangles

2018
*
Discrete Mathematics
*

Triangulations are also known as

doi:10.1016/j.disc.2018.03.018
fatcat:4aqohcb2jnfmblw32bdlw7wmse
*maximal*planar*graphs*. Every triangulation is 3-connected, so by a theorem of Whitney [10] , its embedding is unique. ... Introduction A triangulation shall be a plane*graph*, i.e. a planar*graph*embedded in the plane,*on*at least four vertices, all of whose faces -including the unbounded face -are triangles. ...##
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Structural and computational results on platypus graphs
[article]

2017
*
arXiv
*
pre-print

They are closely related to families of

arXiv:1712.05158v1
fatcat:nez4ywxm3jahvgbsi37p4cpbpq
*graphs*satisfying interesting conditions regarding longest paths and longest cycles, for instance hypohamiltonian, leaf-stable, and*maximally*non-hamiltonian*graphs*... A platypus*graph*is a non-hamiltonian*graph*for which every vertex-deleted subgraph is traceable. ... We would also like to thank Gunnar Brinkmann for his advice*on*plantri. References ...##
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HOMOGENEOUSLY TRACEABLE AND HAMILTONIAN CONNECTED GRAPHS

1984
*
Demonstratio Mathematica
*

*On*the other hand, the join of the trivial

*graph*K.j and a nontrivial HT

*graph*is strongly 1-edge pancyclic. Hypohamiltonian

*graphs*form a subclass of HTNH (HT and

*nonhamiltonian*)

*graphs*. ... Hence each edge of PG is included in exactly

*one*

*maximal*matching of size 3 and two (similar)

*maximal*matchings of size 5. ...

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