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GATE DA 2024 Q2: Minimum colours for a 15-part figure

Question

The 15 parts of the given figure are to be painted such that no two adjacent parts with shared boundaries (excluding corners) have the same color. The minimum number of colors required is

An ellipse containing a square; the square contains a circle split into three sectors, the square's corners are joined to the circle and the ellipse, giving 15 regions
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Answer: A (official GATE 2024 final answer key)

Concept used: Graph colouring of a planar map: regions are vertices, shared boundaries are edges.

First step: Look for a region bordered by a ring of an odd number of regions: the ring itself needs 3 colours, so the centre region needs a 4th.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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