| A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph,where c≥2 is a positive integer.In particular,we call it to be a bipartite tournament when c=2.Let B=(X,Y;E)be a bipartite tournament.We say that B is k-regular if for every vertex v of B,we have dB+(v)=dB-(v)=k.Let F4k be a k-regular bipartite tournament consisting of four independent sets K,L,M,N,in which |K|=|L|=|M|=|N|=k,and all possible arcs are from K to L,from L to M,from M to N,from N to K respectively.Let D be a digraph and let k be a positive integer.A cycle factor of D is a spanning subgraph of D whose components are vertex-disjoint cycles.A cycle factor that contains k cycles is called a k-cycle factor.In 1994,Zhang Kemin,Yannis Manoussakis,and Song Zengmin proposed a conjecture about k-regular bipartite tournaments,that is,any k-regular bipartite tournament that is not isomorphic to F4k has a 2-cycle factor,in which a cycle contains any specific arc and k≥2.The three scholars solved the conjecture that one cycle has the length of 4 and the other has the length of n-4.This article proves the following results.Result 1.Any k-regular bipartite tournament B that is not isomorphic to F4k and k≥3 has a 2-cycle factor of lengths 6 and |V(B)|-6,and one of cycles goes through the arc xy,where xy is any arc of B.Result 2.Let B be a k-regular bipartite tournament that is not isomorphic to F4k and k≥3.Let xy be any arc of B.We prove that B has a cycle C with a length of 6 through xy and B-C is not strongly connected if and only if B has an m-cycle factor,where m≥3.A digraph D is semicomplete if there is at least one arc between every pair of vertices in D.A cycle-tree is either a singleton or consisting of a set of cycles C1,...,Ck such that |V(Ci)∩(V(C1)∪V(Ci-1))|=1 for all i=2,...,k,where V(Cj)is the set of vertices of Cj.When all cycles have the length of three,we speak it a triangle-tree.The mimimum out-degree(resp.minimum in-degree)of D is δ+(D)=min{dD+(x):x∈V(D)}(resp.,δ-(D)=min{dD-(x):x∈V((D)}).The minimum semi-degree of D isδ0(D)=min{δ+(D),δ-(D)}.Frédéric Havet et al.proved that a tournament has aδ+-triangle-tree.This article extends to semicomplete digraph and proves the following result.Result 3.Let D be a strong semicomplete digraph.If δ0(D)≥ 2d and |E(D)|≥n2-2n+2(2 2d-1)+1,then D contains-directed cycles C1,C2,...,Cd such that for all j≤d,|(Ui<jCi)∩Cj|=1,where these cycles are 3 in length. |
