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Detour-saturated graphs of small girths [article]

Pu Qiao, Xingzhi Zhan
2018 arXiv   pre-print
(2) Let Pr be the 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.  ... 
arXiv:1806.06564v1 fatcat:e7gorwdxojg3fi3b44t3lwdmx4

A catalogue of small maximal nonhamiltonian graphs

Janusz Jamrozik, Rafal Kalinowski, Zdzislaw Skupień
1982 Discrete Mathematics  
Special attention is paid to 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.  ... 
doi:10.1016/0012-365x(82)90145-5 fatcat:ryi57476wzfbfiuzgujpjnolna

Maximum nonhamiltonian tough graphs

A. Marczyk, Z. Skupień
1991 Discrete Mathematics  
Skupieri, Maximum 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] .  ... 
doi:10.1016/0012-365x(91)90314-r fatcat:gv4eyany2jc2zbdgr5r5nardpe

Subgraphs intersecting any hamiltonian cycle

P Horák
1988 Journal of combinatorial theory. Series B (Print)  
In terms of the 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  ... 
doi:10.1016/0095-8956(88)90097-4 fatcat:sk3mmjeei5ch7epjn62hkjqgfq

Toughness and nonhamiltonicity of polyhedral graphs

Jochen Harant
1993 Discrete Mathematics  
Tutte the toughness t(G) of a 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.  ... 
doi:10.1016/0012-365x(93)90519-y fatcat:kwk7fodycrdtrbgv3gjfja5w2q

How tough is toughness?

Hajo Broersma
2015 Bulletin of the European Association for Theoretical Computer Science  
We survey results and open problems related to the toughness of 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.  ... 
dblp:journals/eatcs/Broersma15 fatcat:di3kcho5efbktnh73wp4b2ac7e

The cost of perfection for matchings in graphs

E.V. Brazil, C.M.H. de Figueiredo, G.D. da Fonseca, D. Sasaki
2016 Discrete Applied Mathematics  
cubic 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.  ... 
doi:10.1016/j.dam.2014.12.006 fatcat:jmzajjb4r5bnhawz6soavkbgty

The maximum size of a graph with prescribed order, circumference and minimum degree [article]

Leilei Zhang
2021 arXiv   pre-print
All these results 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.  ... 
arXiv:2107.01416v3 fatcat:23rhy432xja2tgrjj5gepwwxoq

Maximal Nontraceable Graphs with Toughness less than One

Frank Bullock, Marietjie Frick, Joy Singleton, Susan Van Aardt, Kieka (C.M.) Mynhardt
2008 Electronic Journal of Combinatorics  
A 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  ... 
doi:10.37236/742 fatcat:k67nq7rv4fca5lc2zgvbr6fd4y

Strongly pancyclic and dual-pancyclic graphs

Terry A. McKee
2009 Discussiones Mathematicae Graph Theory  
Say that a cycle C almost contains a cycle C − if every edge except 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  ... 
doi:10.7151/dmgt.1429 fatcat:ei6hg6lsdfbcpciixymtr26lhq

Planar cubic hypohamiltonian and hypotraceable graphs

Carsten Thomassen
1981 Journal of combinatorial theory. Series B (Print)  
Infinite families of planar cubic hypohamiltonian and hypotraceable 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.  ... 
doi:10.1016/0095-8956(81)90089-7 fatcat:m6maby5libggppn3ileszitv5a

On properties of maximal 1-planar graphs

Dávid Hudák, Tomas Madaras, Yusuke Suzuki
2012 Discussiones Mathematicae Graph Theory  
A 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  ... 
doi:10.7151/dmgt.1639 fatcat:fefldaaa6nhthkowkdpz5f4k7m

Non-hamiltonian triangulations with distant separating triangles

Kenta Ozeki, Carol T. Zamfirescu
2018 Discrete Mathematics  
Triangulations are also known as 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.  ... 
doi:10.1016/j.disc.2018.03.018 fatcat:4aqohcb2jnfmblw32bdlw7wmse

Structural and computational results on platypus graphs [article]

Jan Goedgebeur, Addie Neyt, Carol T. Zamfirescu
2017 arXiv   pre-print
They are closely related to families of 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  ... 
arXiv:1712.05158v1 fatcat:nez4ywxm3jahvgbsi37p4cpbpq

HOMOGENEOUSLY TRACEABLE AND HAMILTONIAN CONNECTED GRAPHS

Zdzislaw Skupień
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.  ... 
doi:10.1515/dema-1984-0424 fatcat:cl6tv3dc6vc2tm2vsvqcx42ziq
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