Question

Problem 8. (2+4+4 points each) A bipartite graph G = (V. E) is a graph whose vertices can be partitioned into two (disjoint) sets V1 and V2, such that every edge joins a vertex in V1 with a vertex in V2. This means no edges are within V1 or V2 (or symbolically: Vu, v E V1. {u, u} &E and Vu, v E V2.{u,v} &E). 8(a) Show that the complete graph K, is a bipartite graph. 8(b) Prove that no complete graph Kn, where n > 2. is a bipartite graph. 8(c) Prove that every rooted tree forms a bipartite graph.

Answer 1

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The Complete graph k, isa bipatrite graph (s)(4) thisis the Complete graph Ky u, since it han two yertices and they are Conneoted by odge. Now o the otharhornd the Verlices of the Complete graph Ka Cembe partionad in two two disjuint set Vi omd V2 where Such Hhont eveny edge (Par ktically 1 edge) is joinigg a edge Veotex in Vi with a vertex in Ve and there arre ho adges with in Vi ond v2 Sa k2 is Complete bipatrite graph. (Proved) Every vouled ree forms a bipatrite graph. We will use the property Ihat eveny Tree T Contains a vertex le of degree 1 amd that T-v is alse a tree. Now we prccedby induction on the numbern of vertiies of a tree. T, for nal, 2 ,K, ond kg arre the only trees ond both are bipartrite - Quppose oy tree on less tham n Vertices s bipertite and Let T be a free n varticess Let v bea vertevof degrce 1 in T omd u beits only

meighbour. we Know that T-v is a tree verlices ,eo by induelion with n-l a ssumpliom , T-v has a bipartitom x, Y. Ihis meams that meng edge in T- hanane end point in X amd other one inY tham X, YUguyis a Now if EX biparti tion oT, Howeverif uEY tipartition of T. then Y, xuu: (fororea) (b) For the Complere graph Kn ny2 of vertey is lo uneeted P air ealh Then if ther.eis a parrtion of vertices like v om d V2 ths Suh Hhat vinve =0 tenen Si Since it isa Coh kn e alh tat p air of memdbers of ri emd Vz but- shonld be lonne eted Sy a nedge thet Contradichs the bipartiZation: so Complete km graph n72 is nota biporrtile araph.