How many colours does a graph with m edges need?

1. Take a proper colouring with exactly $k$ colours and consider two colour classes $C_i,C_j$. If there were no edge between $C_i$ and $C_j$, we could recolour $C_j$ with the colour of $C_i$ and still have a proper colouring with $k-1$ colours, contradicting minimality. So for each of the $\binom k2$ pairs of classes there is an edge joining them, and these edges are distinct. Hence $m\ge\binom k2$, which gives $k\le\dfrac{1+\sqrt{1+8m}}2$.

2. For $m=10$: $k\le\dfrac{1+9}2=5$. The bound is attained by $K_5$, which has $10$ edges and needs $5$ colours.

Remark. Equality $m=\binom k2$ forces exactly one edge between any two classes; $K_k$ is the typical example.

Lesson. In a minimal colouring every pair of colour classes must be linked: use minimality to produce edges.