Bigraphic Pairs With A Realization Containing A Split Bipartite-graph And Turán Number For S_l1∪S_l2

Let Ks,t be the complete bipartite graph with partite sets {x1,...,xs} and {y1,...,yt}.A split bipartite-graph on(s+s′)+(t+t′)vertices,denoted by SBs+s′,t+t′,is the graph obtained from Ks,t by adding s′+t′ new vertices xs+1,...,xs+s′,yt+1,...,yt+t′ such that each of xs+1,...,xs+s′is adjacent to each of y1,...,yt and each of yt+1,...,yt+t’ is adjacent to each of x1,...,xs.Let A and B be nonincreasing lists of nonnegative integers,having lengths m and n,respectively.The pair(A;B)is potentially SBs+s′,t-t′-bigraphic if there is a simple bipartite graph containing SBs+s′,t+t′(with s+s′ vertices x1,...,xs-s′ in the part of size m and t+t′ vertices y1,...,yt-t′in the part of size n)such that the lists of vertex degrees in the two partite sets are A and B.In this paper,we give a characterization for(A;B)to be potentially SBs+s′,t+t′-bigraphic.A simplification of this characterization is also presented.The Turan number of a graph G,denoted by ex(n,G),is the maximum number of edges of an n-vertex simple graph having no G as a subgraph.Let Se denot e the star on l+1 vertices.In this paper,we investigate to determine the Turan number for Sl1∪ Sl2(l1≥l2)and Sl1∪Sl2∪Sl3(l1≥l2≥l3).The results are as follows:(1)We give a new lower bound on ex(n,Sl1∪Sl2):(2)For l2+1≤1≤2l2+1(or l1≥3 and l2=2)and n≥l1+l2+2(or n≥ 2l1+2),we determine the exact values ex(n,Sl1∪Sl2);(3)For l2≥3,l1≥2l2+2 and n≥ 2l1+2l2,we determine the exact values ex(n,Sl1∪Sl2);(4)For l1≥l2≥l3≥ 1 and n≥ max {M1,M2},where both M1 and M2 are two parameters only depending on l1,l2 and l3,we determine the exact values ex(n,Sl1∪ Sl2∪Sl3).The results(1)-(4)improve the corresponding results of Lidicky et al.