Problem 4: For each degree sequence below, determine whether there is a graph with 5 vertices that have these degrees. If a graph exists, draw it. If it doesn't, justify that it doesn't exist. (a) 4, 4, 3, 3, 2. (b) 5, 3, 2, 2, 2. (c) 4, 4, 3, 2, 2. (d) 4, 4, 3, 2, 1.
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- Connect it to another vertex of degree 4. - Connect both of these vertices to a third vertex of degree 3. - Connect the third vertex to two vertices of degree 3 and 2, respectively. Show more…
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Problem 3. (a) For each degree sequence below, determine whether there is a graph with 6 vertices where vertices have these degrees. If a graph exists, draw it. If it doesn't, justify that it doesn't exist. (a1) 4, 4, 4, 3, 3, 1. (a2) 5, 4, 4, 4, 4, 1. (a3) 5, 5, 3, 3, 3, 1. (b) For each degree sequence below, determine whether there is a planar graph with 6 vertices where vertices have these degrees. If a graph exists, draw it. If it doesn't, justify that it doesn't exist. (b1) 5, 5, 4, 4, 3, 3. (b2) 5, 5, 4, 4, 4, 4.
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Exercise 12.7: F For the following degree sequences (list of the degrees of the vertices in a graph) determine if there exists a bipartite simple graph with these degrees. If yes construct it, if not explain why it does not exist. (a) (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2); (b) (4, 4, 4, 4, 4, 3, 3, 2, 2); (c) (5, 5, 5, 5, 5, 5, 4, 4, 4)
(a) For each of the following degree sequences find out whether there is any graph of order 5 with the given degree sequence. If so, give an example (i) 3,3,3,3,2 (ii) 3,3,3,2,2 (b) Up to isomorphism, find all graphs with degree sequence (1, 1, 1, 1, 2, 2, 4). (Hint: the vertex of degree 4 can be adjacent to 0, 1, or 2 of the vertices of degree 2). (c) A graph G has 10 vertices and 21 edges. Find the number of edges in the complement of G.
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