Consider the following coloring questions about complete graphs. (a) Consider K_16, the complete gra ...

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Consider the following coloring questions about complete graphs. (a) Consider K_16, the complete graph with 16 nodes. Each node is colored cardinal, gold, or purple. Prove that there is a node which is the same color as at least 5 of its neighbor nodes. (b) Consider K_{14,16}, the complete bipartite graph with 14 nodes on the left and 16 nodes on the right. Now each edge is colored one of x colors. We want the edges to be colored such that, for any node, all of the edges incident to that node are different colors. What is the minimum value of x necessary? Explain your answer.