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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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The flower conjecture in special classes of graphs

100%
Ryjáček Z., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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1995
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tom 15
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nr 2
179-184
EN
We say that a spanning eulerian subgraph F ⊂ G is a flower in a graph G if there is a vertex u ∈ V(G) (called the center of F) such that all vertices of G except u are of the degree exactly 2 in F. A graph G has the flower property if every vertex of G is a center of a flower. Kaneko conjectured that G has the flower property if and only if G is hamiltonian. In the present paper we prove this conjecture in several special classes of graphs, among others in squares and in a certain subclass of claw-free graphs.
2
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A note on a new condition implying pancyclism

100%
Flandrin E., Li H., Marczyk A., Woźniak M.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 1
137-143
EN
We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n/2,n/2}$.
3
Content available remote

Heavy Subgraphs, Stability and Hamiltonicity

100%
Li B., Ning B.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 3
691-710
EN
Let G be a graph. Adopting the terminology of Broersma et al. and Čada, respectively, we say that G is 2-heavy if every induced claw (K1,3) of G contains two end-vertices each one has degree at least |V (G)|/2; and G is o-heavy if every induced claw of G contains two end-vertices with degree sum at least |V (G)| in G. In this paper, we introduce a new concept, and say that G is S-c-heavy if for a given graph S and every induced subgraph G′ of G isomorphic to S and every maximal clique C of G′, every non-trivial component of G′ − C contains a vertex of degree at least |V (G)|/2 in G. Our original motivation is a theorem of Hu from 1999 that can be stated, in terms of this concept, as every 2-connected 2-heavy and N-c-heavy graph is hamiltonian, where N is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs S such that every 2-connected o-heavy and S-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu’s theorem. Furthermore, our main result improves or extends several previous results.
4
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New sufficient conditions for hamiltonian and pancyclic graphs

100%
Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
29-38
EN
For a graph G of order n we consider the unique partition of its vertex set V(G) = A ∪ B with A = {v ∈ V(G): d(v) ≥ n/2} and B = {v ∈ V(G):d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.
5
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Chvátal-Erdos condition and pancyclism

88%
Flandrin E., Li H., Marczyk A., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
335-342
EN
The well-known Chvátal-Erdős theorem states that if the stability number α of a graph G is not greater than its connectivity then G is hamiltonian. In 1974 Erdős showed that if, additionally, the order of the graph is sufficiently large with respect to α, then G is pancyclic. His proof is based on the properties of cycle-complete graph Ramsey numbers. In this paper we show that a similar result can be easily proved by applying only classical Ramsey numbers.
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