A conjecture on cycle-pancyclism in tournaments

• Volume: 18, Issue: 2, page 243-251
• ISSN: 2083-5892

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Abstract

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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 ${I}_{\gamma }\left(Cₖ\right)=|A\left(\gamma \right)\cap A\left(Cₖ\right)|$, the number of arcs that γ and Cₖ have in common. Let $f\left(k,T,\gamma \right)=max{I}_{\gamma }\left(Cₖ\right)|Cₖ\subset T$ and f(n,k) = minf(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.

How to cite

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Hortensia Galeana-Sánchez, and Sergio Rajsbaum. "A conjecture on cycle-pancyclism in tournaments." Discussiones Mathematicae Graph Theory 18.2 (1998): 243-251. <http://eudml.org/doc/270482>.

@article{HortensiaGaleana1998,
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 $I_γ (Cₖ) = |A(γ)∩A(Cₖ)|$, the number of arcs that γ and Cₖ have in common. Let $f(k,T,γ) = max\{I_γ(Cₖ)|Cₖ ⊂ T\}$ and f(n,k) = minf(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.},
author = {Hortensia Galeana-Sánchez, Sergio Rajsbaum},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Tournaments; pancyclism; cycle-pancyclism; tournament},
language = {eng},
number = {2},
pages = {243-251},
title = {A conjecture on cycle-pancyclism in tournaments},
url = {http://eudml.org/doc/270482},
volume = {18},
year = {1998},
}

TY - JOUR
AU - Hortensia Galeana-Sánchez
AU - Sergio Rajsbaum
TI - A conjecture on cycle-pancyclism in tournaments
JO - Discussiones Mathematicae Graph Theory
PY - 1998
VL - 18
IS - 2
SP - 243
EP - 251
AB - 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 $I_γ (Cₖ) = |A(γ)∩A(Cₖ)|$, the number of arcs that γ and Cₖ have in common. Let $f(k,T,γ) = max{I_γ(Cₖ)|Cₖ ⊂ T}$ and f(n,k) = minf(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.
LA - eng
KW - Tournaments; pancyclism; cycle-pancyclism; tournament
UR - http://eudml.org/doc/270482
ER -

References

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1. [1] B. Alspach, Cycles of each length in regular tournaments, Canadian Math. Bull. 10 (1967) 283-286, doi: 10.4153/CMB-1967-028-6. Zbl0148.43602
2. [2] J. Bang-Jensen and G. Gutin, Paths, Trees and Cycles in Tournaments, Congressus Numer. 115 (1996) 131-170.
3. [3] M. Behzad, G. Chartrand and L. Lesniak-Foster, Graphs & Digraphs (Prindle, Weber & Schmidt International Series, 1979). Zbl0403.05027
4. [4] J.C. Bermond and C. Thomasen, Cycles in digraphs: A survey, J. Graph Theory 5 (1981) 1-43, doi: 10.1002/jgt.3190050102. Zbl0458.05035
5. [5] H. Galeana-Sánchez and S. Rajsbaum, Cycle-Pancyclism in Tournaments I, Graphs and Combinatorics 11 (1995) 233-243, doi: 10.1007/BF01793009. Zbl0833.05039
6. [6] H. Galeana-Sánchez and S. Rajsbaum, Cycle-Pancyclism in Tournaments II, Graphs and Combinatorics 12 (1996) 9-16, doi: 10.1007/BF01858440. Zbl0844.05047
7. [7] H. Galeana-Sánchez and S. Rajsbaum, Cycle-Pancyclism in Tournaments III, Graphs and Combinatorics 13 (1997) 57-63, doi: 10.1007/BF01202236. Zbl0868.05028
8. [8] J.W. Moon, On Subtournaments of a Tournament, Canad. Math. Bull. 9 (1966) 297-301, doi: 10.4153/CMB-1966-038-7. Zbl0141.41204
9. [9] J.W. Moon, Topics on Tournaments (Holt, Rinehart and Winston, New York, 1968). Zbl0191.22701
10. [10] J.W. Moon, On k-cyclic and Pancyclic Arcs in Strong Tournaments, J. Combinatorics, Information and System Sci. 19 (1994) 207-214. Zbl0860.05039
11. [11] D.F. Robinson and L.R. Foulds, Digraphs: Theory and Techniques (Gordon and Breach Science Publishing, 1980). Zbl0484.05034
12. [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.

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