DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 15(2) (1995) 179-184
DOI: 10.7151/dmgt.1015

THE FLOWER CONJECTURE IN SPECIAL CLASSES OF GRAPHS

Zdenk Ryjáček

Department of Mathematics, University of West Bohemia
Americká 42, 306 14 Plze, Czech Republic

Ingo Schiermeyer

Lehrstuhl C für Mathematik, Rhein.n-Westf. Techn. Hochschule
Templergraben 55, D-52062 Aachen, Germany

Abstract

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.

Keywords: hamiltonian graphs, flower conjecture, square, claw-free graphs.

1991 Mathematics Subject Classification: 05C45.

References

[1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).
[2] H. Fleischner, The square of every two-connected graph is hamiltonian, J. Combin. Theory (B) 16 (1974) 29-34, doi: 10.1016/0095-8956(74)90091-4.
[3] H. Fleischner, In the squares of graphs, hamiltonicity and pancyclicity, hamiltonian connectedness and panconnectedness are equivalent concepts, Monatshefte für Math. 82 (1976) 125-149, doi: 10.1007/BF01305995.
[4] A. Kaneko, Research problem, Discrete Math., (to appear).
[5] A. Kaneko and K. Ota, The flower property implies 1-toughness and the existence of a 2-factor, Manuscript (unpublished).