Let G be on a nontrivial connected graph that is not bipartite. Show.pdf

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Let G be on a nontrivial connected graph that is not bipartite. Show that G contains two adjacent vertices u and v such that degu + degv is even. Solution Let V (G) = U ? W, where U = {v ? V (G) : deg v is even} and W = {v ? V (G) : deg v is odd}. Since G is not bipartite, G has at least three vertices, hence max{|U|, |W|} ? 2. Also because G is not bipartite, (U,W) is not a bipartition of G. This implies that U or W contains adjacent vertices u and v. If u, v ? U, then they both have even degree, and if u, v ? W, they both have odd degree. Then degu + degv is even is even..

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