Question

B3 for each integer nzl, define the graph Hn with V (H₂) = {1, 2, ..., }, 4A EZ H.) = { x 5 + V(Ha), xy, q4l xay + n + B. Ê xa anal - adjaceney mohoices. Act) , Actualit at 1 2 3 4 TODO 100 31- oooo ey Suppose that G has n vertices, and for each of i=1,2,..., not that has at least one vertex of degree i. Use induction to prove That G ² Hn.

Answer 1

Proof & By using Induction, with inverting to say se We need to prove that G, N Hn. Base Case & nal single vertex no edge Gi . this and A [hi] = [o]. @ Therefore, G, NH n=2 Gri - Hair Alz=liol Induction Hypothesis & for kan let GR V He, Inductive Step & Considu Gn and Hn. and remove highest degree vertices from both Gn and Hn, respectively This will result into Guy = Gn {v} observe this ) and Han = Hn lv '} le Then, By Induction hypothises Gal{v} s Halfw's Now, add those vertice back in their respective graphs and in the isomorphism from Galers to Halfw's add one more mapping from to ul to get a is emporphism from Gn to Hn. Gn Therefore, I Hn. le GN Hn. where Gas a graph with a vertices & given property.