An approximate version of Jackson’s conjecture

A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

A Note on the Feedback Arc Set Problem and Acyclic Subdigraphs in Bipartite Tournaments

Given a digraph D a feedback arc set is a subset X of the arcs of D such that D − X is acyclic. Let β(D) denote de minimum cardinality of a feedback arc set of D. In this paper we prove that a bipartite tournament T with minimum out-degree at least r satisfies β(T) ≥ r2. A lower bound and an upper bound for β(T) are given in terms of the bipartite dichromatic number. We define the bipartite dichromatic number of a balanced bipartite tournament Tn,n and use this invariant to give an upper bound for the minimum cardinality of a feedback arc set of Tn,n. We also prove that for each positive integer k ≥ 3 there is an integer N(k) such that if n ≥ N(k), then each balanced bipartite tournament contains an acyclic bipartite tournament Tk,k.

Finding a Longest Alternating Cycle in a 2-edge-coloured Complete Graph is in RP

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

On the Distribution of 4-Cycles in Random Bipartite Tournaments

Let there be given two sets of points, P = {P1,…, Pm}1 m and Q={Q1,…, Qn}, such that joining each pair of points (Pi, Qk), for i=1,…, m and k=1…, n, is a line oriented towards one, and only one, of the pair. Such a configuration will be called an m×n bipartite tournament. If the line joining Pi to Qk is oriented towards Qk we may indicate this by pi→Qk, and similarly if the line is oriented in the other sense. The points Pi, Pj, Qk, and Ql. will be said to form a 4-cycle if either Pi→Qk→Pj→Ql→Pi or Pi→Ql→Pj→Qk→Pi. C(m, n), the number of 4-cycles in a given m×n bipartite tournament, provides, in some sense, a measure of the degree of transitivity of the relationship indicated by the orientation of the lines, and the complete configuration may be thought of as representing the outcome of comparing each member of one population with each member of a second population, and making a decision, upon some basis, as to which component of each pair is the preferred one.