. . , vn}, E) is defined as (deg(v1), deg(v 2), . . . , deg(vn)) (i.e. it is a sequence containing the degree of each vertex in some order). Show that there exists no graph G = (V, E) with |V | = 12 and degree sequence (1, 1, 2, 2, 3, 4, 6, 6, 7, 7, 8, 8). 2) Prove that if G is not connected then G is connected. 3) Let G be a graph with n vertices and k connected components. What is the largest number of edges that G could contain? 4) Prove that if a connected graph has exactly two vertices having odd degree, then it has an Eulerian walk whose endpoints are the vertices with odd degree. (Hint: Use the fact that a connected graph with only even degree vertices contains a closed Eulerian walk.) 5) Consider the complete graphs Kn for n _ 1. a) For what values of n does Kn have an Eulerian Walk (the walk can be either closed or not closed)? b) For what values of n does Kn have a closed Eulerian walk? c) For what values of n does Kn contain an Eulerian walk that is not closed? (You can assume the fact that any connected graph with more than two odd vertices does not have an Eulerian walk.)
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