Question

Show that the Grotzsch graph is non-planer.
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Answer 1

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Answer 2

Explanation:

Non-Planar Graph:

A graph is said to be non planar if it cannot be drawn in a plane so that no edge cross.

Example: The graphs shown in fig are non planar graphs.

Planar and Non-Planar Graphs

These graphs cannot be drawn in a plane so that no edges cross hence they are non-planar graphs.

Properties of Non-Planar Graphs:

A graph is non-planar if and only if it contains a subgraph homeomorphic to K5 or K3,3

Example1: Show that K5 is non-planar.

Solution: The complete graph K5 contains 5 vertices and 10 edges.

Now, for a connected planar graph 3v-e≥6.

Hence, for K5, we have 3 x 5-10=5 (which does not satisfy property 3 because it must be greater than or equal to 6).

Thus, K5 is a non-planar graph.