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- At a cupcake stall at the Greenwich market, the owner is selling six different types of cupcakes. The owner sells them in boxes where each box contains exactly two cupcakes, each of different type. Each type of cupcake is used in combination with at least three others. (a) Let G be a graph where each vertex represents one of the cupcakes and where an edge joins two vertices whenever two cupcakes are used in the same box. Let v be an arbitrary vertex in G. What is the smallest degree v can have? Justify your answer. ' ^ (b) Show that there are three gift boxes which between them have all six types of cupcakes.Prove that a graph is two-colorable (bipartite) if and only if it contains no odd-length cycle.Prove that for any two edges of a 2-connected graph, a cycle exists containing both of them.
- b.) Determine whether the graph is simple or not. If it is a simple the adjacency matrix of the graph and determine the no. of paths of length 2 that it has.Prove that every Cayley graph is vertex-transitive.Show that any graph with two or more nodes and no self-loops contains two nodes that have equal degrees.
- If G = (V, E) has n > 2 vertices and no self-loops, show that there exist two vertices v # w such that deg(v) = deg(w). Present a counterexample, if G is allowed to have self-loops.Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.Prove that the dual to Eulerian planar graph is bipartite.