Long cycles in bipartite tournaments

On the existence of specified cycles in bipartite tournaments

For two integers n ≥ 3 and 2 ≤ p ≤ n, we denote D(n, p) the digraph obtained from a directed n-cycle by changing the orientations of p − 1 consecutive arcs. In this paper, we show that a family of k-regular (k ≥ 3) bipartite tournament BT4k contains D(4k, p) for all 2 ≤ p ≤ 4k unless BT4k is isomorphic to a digraph D such that (1, 2, 3, ..., 4k, 1) is a Hamilton cycle and (4m+ i− 1, i) ∈ A(D) a...

Complementary cycles containing a fixed arc in diregular bipartite tournaments

Let (x, y) be a specified arc in a k-regular bipartite tournament B. We prove that there exists a cycle C of length four through (x, y) in B such that B-C is hamiltonian.

A sufficient condition for Hamiltonian cycles in bipartite tournaments

We prove a new sufficient conditi()n on degrees for a bipartite tournament to be Hamiltonian, that is, if an n x n bipartite tournament T satisfies the condition dT(u) + dj;(v) ~ n 1 whenever uv is an arc of T, then T is Hamiltonian, except for two exceptional graphs. This result is shown to be best possible in a sense. T(X, Y, E) denotes a bipartite tournament with bipartition (X, Y) and verte...

A sufficient condition for Hamilton cycles in bipartite tournaments

Long cycles in unbalanced bipartite graphs

Let G[X,Y ] be a 2-connected bipartite graph with |X| ≥ |Y |. For S ⊆ V (G), we define δ(S;G) := min{dG(v) : v ∈ S}. We define σ1,1(G) := min{dG(x) + dG(y) : x ∈ X, y ∈ Y, xy / ∈ E(G)} and σ2(X) := min{dG(x) + dG(y) : x, y ∈ X,x 6= y}. We denote by c(G) the length of a longest cycle in G. Jackson [J. Combin. Theory Ser. B 38 (1985), 118–131] proved that c(G) ≥ min{2δ(X;G) + 2δ(Y ;G)− 2, 4δ(X;G)...