2. If the simple graph G has n vertices and m edges, how many edges does...
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2. If the simple graph G has n vertices and m edges, how many edges does G have? 3. Suppose that graph G is isomorphic to graph G. Show that if G contains a cycle of length k so does G'. 4. Prove that, if two different cycles of a graph G each contain an edge e, then G has a cycle that doesn't contain e. 5. Show that two graphs G and H are isomorphic if and only if their complements G and H are isomorphic. 6. Let G be a k-regular bipartite graph (k21), show that the two partitions have the same size. 7. A bridge of a connected graph G is an edge e such that G-e is disconnected. Prove that if e is a bridge, then e doesn't lie of any cycle in G. 8. Let G be a graph where all vertices have degree at least 8, for some 82 2. Prove that G contains a cycle of length at least 8 + 1. 9. A simple graph that is isomorphic to its complement is called self-complementary. Prove that, if the graph G is self-complementary, then G has 4k or 4k+1 vertices, where k is an integer. 10. Which of the following pairs of graphs are isomorphic? Explain carefully. (a) (b) g g a h d 6 8 4 5 2 7 4 3 LO 3 5 8 2. If the simple graph G has n vertices and m edges, how many edges does G have? 3. Suppose that graph G is isomorphic to graph G. Show that if G contains a cycle of length k so does G'. 4. Prove that, if two different cycles of a graph G each contain an edge e, then G has a cycle that doesn't contain e. 5. Show that two graphs G and H are isomorphic if and only if their complements G and H are isomorphic. 6. Let G be a k-regular bipartite graph (k21), show that the two partitions have the same size. 7. A bridge of a connected graph G is an edge e such that G-e is disconnected. Prove that if e is a bridge, then e doesn't lie of any cycle in G. 8. Let G be a graph where all vertices have degree at least 8, for some 82 2. Prove that G contains a cycle of length at least 8 + 1. 9. A simple graph that is isomorphic to its complement is called self-complementary. Prove that, if the graph G is self-complementary, then G has 4k or 4k+1 vertices, where k is an integer. 10. Which of the following pairs of graphs are isomorphic? Explain carefully. (a) (b) g g a h d 6 8 4 5 2 7 4 3 LO 3 5 8
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1 Answer A spanning tree is a subgraph of a graph that contains all the vertices of the graph and is a tree Spanning trees are useful for solving many problems in graph theory such as finding minimal ... View the full answer
Related Book For
Discrete and Combinatorial Mathematics An Applied Introduction
ISBN: 978-0201726343
5th edition
Authors: Ralph P. Grimaldi
Posted Date:
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