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)

On a cyclic connectivity property of directed graphs

AbstractLet us call a digraph D cycle-connected if for every pair of vertices u,v∈V(D) there exists a cycle containing both u and v. In this paper we study the following open problem introduced by Ádám. Let D be a cycle-connected digraph. Does there exist a universal edge in D, i.e., an edge e∈E(D) such that for every w∈V(D) there exists a cycle C such that w∈V(C) and e∈E(C)?In his 2001 paper Hetyei conjectured that cycle-connectivity always implies the existence of a universal edge. In the present paper we prove the conjecture of Hetyei for bitournaments

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 [15] (see also [14]). 1 Introduction A semicomplete multipartite digraph is a digraph D = (V (D); A(D)) for which the vertices of V (D) can be partitioned into a number k 2 of subsets (called colour classes) such that every pair of vertices from the same colour class are non-adjacent and every pair of vertices from different colour classes are adjacent (i.e. there is at least one arc between them). Two well-known special cases are semicomplete digraphs (when k = jV (D)j) and semicomplete bipartite digraphs (when k = 2). It is well-known that every strongly connected semicomplete digraph is Hamiltonian [9] and that a Hamiltonian cycle in a strong semicomplete digraph on n vertices can be found in time O(n 2 ) [17]. For semicomplete bipartite digraphs strong connectivity by itself is not enough to guarantee the existence o..