Chapter 10: Q61SE (page 738)
Show that the number of vertices in a simple graph is less than or equal to the product of the independence number and the chromatic number of the graph.
Most of the ki vertices are proving the required thing.
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Show that if \({\bf{G}}\) is a simple graph with at least 11 vertices, then either \({\bf{G}}\)or\({\bf{\bar G}}\), the complement of \({\bf{G}}\), is non-planar.
A tournament is a simple directed graph such that if u and v are distinct vertices in the graph, exactly one of (u, v) and (v, u) is an edge of the graph.Show that every tournament has a Hamilton path.
Solve the art gallery problem by proving the art gallery theorem, which states that at most (n/3) guards are needed to guard the interior and boundary of a simple polygon with n vertices.
(Hint: Use Theorem 1 in Section 5.2 to triangulate the simple polygon into n - 2 triangles. Then show that it is possible to color the vertices of the triangulated polygon using three colors so that no two adjacent vertices have the same color. Use induction and Exercise 23 in Section 5.2. Finally, put guards at all vertices that are colored red, where red is the color used least in the coloring of the vertices. Show that placing guards at these points is all that is needed.)
Explain how to find a path with the least number of edges between two vertices in an undirected graph by considering it as a shortest path problem in a weighted graph.
Show that a connected graph with optimal connectivity must be regular.
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