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1
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On theH-Force Number of Hamiltonian Graphs and Cycle Extendability

100%
Hexel E.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
79-88
EN
The H-force number h(G) of a hamiltonian graph G is the smallest cardinality of a set A ⊆ V (G) such that each cycle containing all vertices of A is hamiltonian. In this paper a lower and an upper bound of h(G) is given. Such graphs, for which h(G) assumes the lower bound are characterized by a cycle extendability property. The H-force number of hamiltonian graphs which are exactly 2-connected can be calculated by a decomposition formula.
2
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Forbidden Subgraphs for Hamiltonicity of 1-Tough Graphs

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Li B., Broersma H. J., Zhang S.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
|
nr 4
915-929
EN
A graph G is said to be 1-tough if for every vertex cut S of G, the number of components of G − S does not exceed |S|. Being 1-tough is an obvious necessary condition for a graph to be hamiltonian, but it is not sufficient in general. We study the problem of characterizing all graphs H such that every 1-tough H-free graph is hamiltonian. We almost obtain a complete solution to this problem, leaving H = K1 ∪ P4 as the only open case.
3
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Extension of several sufficient conditions for Hamiltonian graphs

100%
Ainouche A.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 1
23-39
EN
Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖{u})|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_r ∪ Kₛ ∪ Kₜ) ∨ K₂$, 1 ≤ r ≤ s ≤ t or the graph $((n+1)/2)K₁ ∨ K_{(n-1)/2}$. It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.
4
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A classification for maximal nonhamiltonian Burkard-Hammer graphs

88%
Tan N., Iamjaroen C.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 1
67-89
EN
A graph G = (V,E) is called a split graph if there exists a partition V = I∪K such that the subgraphs G[I] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I| < |K| to be hamiltonian. We will call a split graph G with |I| < |K| satisfying this condition a Burkard-Hammer graph. Further, a split graph G is called a maximal nonhamiltonian split graph if G is nonhamiltonian but G+uv is hamiltonian for every uv ∉ E where u ∈ I and v ∈ K. Recently, Ngo Dac Tan and Le Xuan Hung have classified maximal nonhamiltonian Burkard-Hammer graphs G with minimum degree δ(G) ≥ |I|- 3. In this paper, we classify maximal nonhamiltonian Burkard-Hammer graphs G with |I| ≠ 6,7 and δ(G) = |I| - 4.
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