1. Show that for every *connected* multigraph G on n>1 vertices, it is possible to add at most floor(n/2) edges to G so that the resulting graph has an Euler circuit. (hint: use the handshaking lemma) 2. Prove that a simple graph G=(V,E) has a k-coloring if and only if there is a partition of its vertices into subsets V_1,...,V_k such that for all i=1...k, no two vertices in V_i are adjacent in G. 3. The complement of a simple graph G=(V,E) is the graph G'=(V,{ {u,v} : {u,v} is not an element of E} }, i.e., the graph whose edges are the non-edges of G. Show that for every graph G, if G is not connected then its complement must be connected.