proof this Show that K, is not planar.

Name: proof this Show that K, is not planar.
Uploaded: 2024-03-20T06:38:44.000Z
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Description: proof this Show that K, is not planar.

Answered: Show that K, is not planar.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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proof this

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Step 1

To prove: $K_{5}$ is not a planar graph.

Proof:

This is proved by the method of contradiction.

We use two theorems to prove this and the result follows immediately from them.

Let $G = (V, E)$ be a connected planar graph and let $v = | V |$ , $e = | E |$ and $r$ be the number of regions in which some given embedding of $G$ divides the plane.

1. Euler's formula: $v - e + r = 2$

2. If $e \geq 3$ then $3 r \leq 2 e$ and $e \leq 3 v - 6$

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