A conjecture on cycle-pancyclism in tournaments

• Autorzy: Hortensia Galeana-Sánchez , Sergio Rajsbaum
• Języki publikacji: EN

Abstrakty

EN

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) = 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.

Słowa kluczowe

EN

Tournaments; pancyclism; cycle-pancyclism

Kategorie tematyczne

• 05C20: Directed graphs (digraphs), tournaments

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1998
• Tom: 18
• Numer: 2
• Strony: 243-251

Daty

wydano

1998

otrzymano

1998-09-28

Twórcy

autor

Hortensia Galeana-Sánchez

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

autor

Sergio Rajsbaum

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

Bibliografia

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

Identyfikatory

DOI

10.7151/dmgt.1080