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Discrete Mathematics and its Applications
Found in: Page 738
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

To show the expression \(e \le 2v - 4{\rm{ if }}v \ge 3\).

The expression is shown \(e \le 2v - 4{\rm{ if }}v \ge 3\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1:  Given

A connected bipartite planar simple graph has e edges and v vertices.\(v \ge 3\)

Step 2: The Concept ofbipartite graph

Abipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and, that is every edge connects a vertex in to onein.

Step 3: Determine the expression

Let the vertices of \({\rm{G}}\) be partitioned into two sets \({V_1}\) and \({V_2}\)

A bipartite graph can only have circuits of even length, because if we have a vertex in \({V_1}\), then we need to use an even number of edges to end up at a vertex in \({V_1}\) again.

Then a bipartite graph does not have a circuit of length 3 and we know that \(e \le 2v - 4\) by corollary.

Most popular questions for Math Textbooks

a) Show that if the diameter of the simple graph \({\bf{G}}\) is at least four, then the diameter of its complement \({\bf{\bar G}}\) is no more than two.

b) Show that if the diameter of the simple graph\({\bf{G}}\) is at least three, then the diameter of its complement \({\bf{\bar G}}\) is no more than three.

The distance between two distinct vertices \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\)of a connected simple graph is the length (number of edges) of the shortest path between \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\). The radius of a graph is the minimum over all vertices \({\bf{v}}\) of the maximum distance from \({\bf{v}}\) to another vertex. The diameter of a graph is the maximum distance between two distinct vertices. Find the radius and diameter of

a) \({{\bf{K}}_{\bf{6}}}\)

b) \({{\bf{K}}_{{\bf{4,5}}}}\)

c) \({{\bf{Q}}_{\bf{3}}}\)

d) \({{\bf{C}}_{\bf{6}}}\)

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.)

Extend Dijkstra’s algorithm for finding the length of a shortest path between two vertices in a weighted simple connected graph so that a shortest path between these vertices is constructed.

Determine whether the given simple graph is orientable.
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