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The flower conjecture in special classes of graphs

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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.
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On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs

100%
Seamone B.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 2
207-214
EN
A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.
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On the Erdős-Gyárfás Conjecture in Claw-Free Graphs

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Nowbandegani P. S., Esfandiari H., Haghighi M. H. S., Bibak K.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 3
635-640
EN
The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs
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