Question

(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G.

Answer 1

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plane araph with edn and faces. Aud k (G) iA He numben ofComnoctd compononts G Nots, lat ae havivg k componemts, bay G vertices, ut ni,mifi be number J Ventices, ader and faces in Gi 1ei sk. Eulenls hormula is applicable for each componemt , So, F e 1LiK By Bumming au Hhase, ni-mitfi-2, K i fi (K-L) 2k, be cause we Computin taces compovants each time We Compute Hhe outen face o G and G is having So hor G, We get one outer face only Y mtfk+ m-mt K(G)+ 1. CsScanned with CamScanner