Chapter 9 261 5. (a) JC4, the complete graph on 4 vertices has this degree sequence. (b) No such graph exists because the sum of the degrees in a graph must be even, but the sum here is 15, which is not even. (c) Draw two triangles, with vertices AI, Bl, Cl and A2, B2, C2, respectively. Then add edges AIA2, BIB2, CIC2 . This gives a graph with the given degree sequence. (d) Drawing a graph by the approach of part (c), but starting with two squares, gives a graph with the given degree sequence. 6. (a) The sum of the degrees is 41, an odd number, contradicting Proposition 9.2.5. (b) Suppose such a graph existed. It would have nine vertices two of which have degree 8. Hence, these two would be joined to all vertices except themselves, and thus every vertex would have degree at least 2. This contradicts the fact that there is a vertex of degree 1. 7. (a) This is the complete bipartite graph JC3,4' (b) This is not bipartite. It contains an odd cycle (the vertices along the top and right side). (c) This is bipartite, but not complete. There are black and white vertices not joined by an edge.
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