Hamiltonian Of Hypertournaments And Bipartite Hypertournamnets

As a branch of graph theory,hypergraphs play a special role in graph theory and are powerful tools for studying discrete problems.Hypertournament and bipartite hypertournament are an interesting research topic in graph theory.They are the popularization of general tournament and bipartite tournament,respectively,which have good research value and practical application prospect.Long before,some well-known expects and scholars such as,Barbut,Bialostcki and Frankl did research on hypertournaments.They put forward an important conclusion that each hypertournament with n ? k + 1 ? 2 contains a Hamiltonian path and every strong hypertournament with n ? k + 2 ? 3 contains a Hamiltonian cycle.Petrovic and Yang gave a sufficient condition for a k-hypertournament is d-disjoint-vertex-pancyclic,respectively.In this paper,we mainly study the Hamiltonian of hypertournament and bipartite hypertournamnet,as follows:Chapter one,the application background,the development of hypergrpahs and basic concepts are introduced.And the contents of this paper is proposed.Chapter two,we discuss the d-disjoint-vertex-pancyclicity of k-hypertournaments and extend results of Petrovic and Yang.The following conclusion is obtained:Let T be a k-hypertournament on n vertices and d be an integer with d ? 1.If one of the following holds:(i)k ? 5 and n ? 2d + k;(ii)k = 4 and n ? 5d + 1;(iii)k = 3 and n ? 8d-3,then T is d-disjoint-vertex-pancyclic if and only if T is d-arc-connected.Chapter three,we charaterize the majority digraph of [h,k]-bipartite hypertournaments and study Hamiltonian of [h,k]-bipartite hypertournament with bipartition(U,W),where |U | = n and|W | = m.And the following results are obtained:1.Every [h,k]-bipartite hypertournament BT with m + n vertices,where 2 ? h ? m-1 and2 ? k ? n-1,contains a Hamiltonian path.2.Let BT be a strong [h,k]-bipartite hypertournament on m + n vertices,where m ? h + 2 ? 4and n ? k + 2 ? 4.If one of the following holds:(i)k = 2 and h ? 3;(ii)k ? 3 and h = 2;(iii)k ? 3 and h ? 3,then BT has a Hamiltonian cycle.Chapter four,the summary of this paper is given.