Discussiones Mathematicae Graph Theory 15(2) (1995)
179-184
DOI: https://doi.org/10.7151/dmgt.1015
THE FLOWER CONJECTURE IN SPECIAL CLASSES OF GRAPHS
Zdenk Ryjáček Department of Mathematics, University of West Bohemia |
Ingo Schiermeyer Lehrstuhl C für Mathematik, Rhein.n-Westf. Techn.
Hochschule |
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[5] | A. Kaneko and K. Ota, The flower property implies 1-toughness and the existence of a 2-factor, Manuscript (unpublished). |
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