Hamiltonian paths containing a given arc, in almost regular bipartite tournaments

AbstractA tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d+(x) and d−(x) the outdegree and indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D)=max{d+(x),d−(x)}−min{d+(y),d−(y)} over all vertices x and y of D (including x=y). If ig(D)⩽1, then D is called almost regular, and if ig(D)=0, then D is regular.More than 10 years ago, Amar and Manoussakis and independently Wang proved that every arc of a regular bipartite tournament is contained in a directed Hamiltonian cycle. In this paper, we prove that every arc of an almost regular bipartite tournament T is contained in a directed Hamiltonian path if and only if the cardinalities of the partite sets differ by at most one and T is not isomorphic to T3,3, where T3,3 is an almost regular bipartite tournament with three vertices in each partite set.As an application of this theorem and other results, we show that every arc of an almost regular c-partite tournament D with the partite sets V1,V2,…,Vc such that |V1|=|V2|=⋯=|Vc|, is contained in a directed Hamiltonian path if and only if D is not isomorphic to T3,3

A polynomial algorithm for the Hamiltonian cycle problem in semicomplete multipartite digraphs

We describe a polynomial algorithm for the Hamiltonian cycle problem for semicomplete multipartite digraphs. The existence of such an algorithm was conjectured in G. Gutin, Paths and cycles in digraphs. Ph. D. thesis, Tel Aviv Univ., 1993. (see also G. Gutin, J Graph Theory 19 (1995) 481-505)

Vertex heaviest paths and cycles in quasi-transitive digraphs.

A digraph D is called a quasi-transitive digraph (QTD) if for any triple x,y,z of distinct vertices of D such that (x, y) and (y, z) are arcs of D there is at least one arc from x and z or from z to x. Solving a conjecture by Bang-Jensen and Huang (1995), Gutin (1995) described polynomial algorithms for finding a Hamiltonian cycle and a Hamiltonian path (if it exists) in a QTD. The approach taken in that paper cannot be used to find a longest path or cycle in polynomial time. We present a principally new approach that leads to polynomial algorithms for finding vertex heaviest paths and cycles in QTDs with non-negative weights on the vertices. This, in particular, provides an answer to a question by N. Alon on longest paths and cycles in QTDs