3. In this MATH 163 problem, we invite you to stare at this beautiful graph. 2 Your task is to simply explore the crap out of this graph, and record your findings. Is it con- nected? What's the shortest/longest path/cycle in the graph? What's its chromatic number? Noteworthy alternative drawings of the graph? What other interesting questions can you come up with about this graph (even if you might not be able to fully answer them)? All submissions that exhibit some effort will receive full credit. Again, you decide how much you get out of this exercise. 4. In this MATH 163 problem, we investigate if there exist graphs with certain combination of properties. For each of the following, either give an example of a graph with the given properties, or explain why no such graph exists. (a) A 7-vertex graph in which all vertices have the same degree. (b) A 7-vertex graph in which no two vertices have the same degree. (c) A 7-vertex graph with 6 edges, such that every vertex is contained in a cycle. 5. In this MATH 163 assignment problem, we investigate the complement of cycles. Given a graph G = (VG, EG), its complement graph G = (Va. EG) is defined so that . Va = VG (i.e., G and G have the same set of vertices). . Two vertices i, j E Va are adjacent in G if and only if they are not adjacent in G. Also, for every n E N where n 2 3, let C, denote the graph that is the n-cycle. For example, the following is the graph C's. (a) Draw the graphs C3, C4, Cs, and Ce. (b) In general, how many edges are there in C,? Explain your answer. 6. In this MATH 163 assignment problem, we propose the following definition: Definition. Given a graph G = (VG, Ec) that is connected, we say that G is super-connected if, for every verter v E Va, removing v and all edges incident with v from G results in a connected graph. (a) Determine if each of the following graphs is connected and/or super-connected. Briefly justify your responses. G1 G2 G3 (b) Prove that every tree on at least three vertices is not super-connected