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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

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.

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