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The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let G be any graph with n vertices and chromatic number k. Which of the following statements is/are always TRUE?

1. G contains a complete subgraph with k vertices
2. G contains an independent set of size at least n/k
3. G contains at least k(k − 1)/2 edges
4. G contains a vertex of degree at least k

Correct answer: G contains an independent set of size at least n/k

Solution

A graph with chromatic number k can be colored using k colors, which implies that there must be at least one independent set of vertices that can be colored with the same color. By the pigeonhole principle, if there are n vertices and k colors, at least one color must be assigned to at least n/k vertices, ensuring the existence of an independent set of that size.

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