Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h(k)}$ of length h(k), h(k) ∈ {k,k-2} with $|A(C_{h(k)}) ∩ A(γ)| ≥ h(k)-3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k < [(n+4)/2] will be studied.
Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h(k)}$ of length h(k), h(k) ∈ {k,k-2} with $|A(C_{h(k)}) ∩ A(γ)| ≥ h(k)-4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was proved.
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.
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