**Question 4: Planarity of $ K_{3,3} $**In answering the following questions, you will demonstrate that $ K_{3,3} $ is not planar. Consider, for the sake of contradiction, that $ K_{3,3} $ is planar and has a planar representation.**(a)** Since $ K_{3,3} $ is assumed to have a planar representation, $ K_{3,3} $ has faces. How many faces does $ K_{3,3} $ have? Explain.**(b)** $ K_{3,3} $ will have no face of length 3. Why?**(c)** Based on your answer to (b), $ K_{3,3} $ has at least how many edges? Explain.

Answered: 1. In answering the following questions…

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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In the question first, we have to show whether $K_{3, 3}$ is a planar graph or not. Also, it is asked it has how many faces. Also it shall have no face of length $3 .$

Also, we have to show it has how many edges. First we shall know what a $k_{3, 3}$ is.

$k_{3, 3}$ has $6$ vertices and $9$ edges. A graph $G$ is planar if it can be drawn in the plane with its edges intersecting at their vertices only. One such graph is called an embedding graph of the line. A particular planar representation of a planar graph is called a map. A map divides the plane into a number of region or planes ( one of them infinite).