Almost everywhere I read it defines chromatic number as the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color. But if adjacent vertices are of different colours, it seems to mean that adjacent vertices belong to different independent sets. So, I'm almost certain that the definition in title is correct, but I couldn't find a standard source where it is defined like it. So, I'm posting this question to get confirmation.
Isn’t chromatic number just the minimum number of independent sets
coloringdiscrete mathematicsgraph theory
Best Answer
Yes. More specifically, the chromatic number of $G=(V,E)$ is the minimum size of a partition of $V$ such that each part of the partition is an independent set.
Given a partition into $k$ independent sets, you can get a $k$-colouring by colouring all the vertices in part $P_i\subset V$ the colour $i$.
And, given a $k$-colouring $c: V\to \{1,2,\dots, k\}$, the colour classes $c^{-1}(i)$ form a partition of $V$ into $k$ independent sets.