The Three Results Of Bipartite Tournaments

Bipartite tournaments are a special class of digraphs.There are many conclusions on bipartite tournaments.In this thesis,we mainly given some conclusions of bipartite tournaments:For arc-coloured bipartite tournaments,we give some sufficient conditions for the existence of kernels by rainbow paths,and calculate the lower bound of properly coloured 4-cycles in a strong arc-coloured bipartite tournament;For bipartite tournaments without arc-coloured,we characterized the domination graph of a bipartite tournament and calculate some related parameters of the domination graph.The relation between the domination graph and the competition graph of a digraph is given.For convenience,let D(X,Y)be a bipartite tournament,X,Y be two partite sets of D.The thesis consists of four sections.Chapter 1 is the preface.The basic concepts are introduced.And the application background of main content is proposed.In Chapter 2,firstly,we studied the existence of kernels by rainbow paths in arc-colored bipartite tournaments by using the perfection of rainbow closures of arc-colored bipartite tournaments.And proved that when min{|X|,|Y|}=2,if all 4-cycles contained in D are coloured with 3 colours or when min{|X|,|y|}?3,if all 4-cycles,6-cycles and induced subdigraphs CB5 are rainbow,and all induced subdigraphs TB4 are properly coloured,then the bipartite tournament has kernels by rainbow paths.In addition,by same method,we studied the kernels by rainbow paths in other arc-coloured digraphs and proved that:Unicyclic digraphs with the unique cycle rainbow have a RP-kernel;Semicomplete digraphs with all 3-cycles rainbow have a one-vertex RP-kernel;Quasi-transitive digraphs with all 3-cycles and all induced subdigraphs Q4 rainbow have a RP-kernel.In Chapter 3,we get that the lower bound of properly coloured 4-cycles passing through the arc(x,y)in a strong arc-coloured bipartite tournament is:?[xy)(2?(xy)-|X|-i(D)+1/2-(?+mond+(y)+?-mond-(x)+?+mon?-mon).where ?(xy)=min{d+(y),d-(x)},i(D)=maxv?(D)|d+(v)-d-(v)|,?+mon(?-mon)is the maximum out(in)monochromatic degree.In Chapter 4,firstly,we characterize the domination graph dom(D)of a bipartite tournament and prove that dom(D)is the union of a star and some disjoint vertices,except when |X|=|Y|=2,dom(D)might be the union of two stars.In addition,we calculate some related parameters of the domination graph and obtains the following:?(dom(D))is either |V(D)|,or |V(D)|-1,or |V(D)|-2;cc(dom(D))is either |V(D)|,or |V(D)|-1,or|V(D)|—2;?(dom(D))is either 1 or 2;?(dom(D))<max{|X|,|Y|}+1.We also get the relation between the domination graph and the competition graph of a digraph.