DMGT

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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 18(2) (1998) 243-251
DOI: https://doi.org/10.7151/dmgt.1080

A CONJECTURE ON CYCLE-PANCYCLISM  IN TOURNAMENTS

Hortensia Galeana-Sánchez and Sergio Rajsbaum

Instituto de Matemáticas, U.N.A.M.
C.U., Circuito Exterior, D.F. 04510, México

e-mail: rajsbaum@servidor.unam.mx

Abstract

Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Ck of length k in T we denote Iγ (Ck) = |A(γ)∩A(Ck) |, the number of arcs that γ and Ck have in common. Let f(k,T,γ) = max{ Iγ(Ck)|Ck ⊂ T} and f(n,k) = min{ f(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T}. In previous papers we gave a characterization of f(n,k). In particular, the characterization implies that f(n,k) ≥ k-4.

The purpose of this paper is to give some support to the following original conjecture: for any vertex v there exists a cycle of length k containing v with f(n,k) arcs in common with γ.

Keywords: Tournaments, pancyclism, cycle-pancyclism.

1991 Mathematics Subject Classification: 05C20.

References

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[2] J. Bang-Jensen and G. Gutin, Paths, Trees and Cycles in Tournaments, Congressus Numer. 115 (1996) 131-170.
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[9] J.W. Moon, Topics on Tournaments (Holt, Rinehart and Winston, New York, 1968).
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[12] Z.-S. Wu, k.-M. Zhang and Y. Zou, A Necessary and Sufficient Condition for Arc-pancyclicity of Tournaments, Sci. Sinica 8 (1981) 915-919.

Received 28 September 1998

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