I am stuck with the following exercise:
Let $\sigma$ be an ordering of the vertices of a given graph $G$ and let $\chi(G,\sigma)$ be the number of colours the greedy colouring algorithm uses. Show that
$$\chi(G,\sigma) \leq \frac{1}{2} + \sqrt{2E(G)+\frac{1}{4}}$$
I know that for the chromatic number of any graph holds $\chi(G) \leq \sqrt{2E(G)+\frac{1}{4}}$, but I do not see why this should hold for $\chi(G,\sigma)$ too. Could you help me?