How Many Colors Does Any Bipartite Graph Need? | Graph Theory Trick

Chromatic number sounds complicated until you see what bipartite actually means — and then the answer becomes obvious.
Every bipartite graph has two sides. Every edge crosses between them. So paint one side red, the other blue, and you're done — no two neighbours ever share a colour. It doesn't matter how many vertices there are, how many edges there are, or how tangled the connections look. Two colours always work and one never does.
The deeper result is even more satisfying: a graph needs at most 2 colours if and only if it contains no odd cycle. Bipartite graphs have only even cycles — that's the structural reason this works.

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