A first moment proof of the Johansson-Molloy theorem

We give a short proof of a stronger form of the Johansson-Molloy theorem which relies only on the first moment method. Our result implies that, for every graph $G$ of maximum degree $\Delta$ where every neighbourhood induces a subgraph of maximum degree $d$, $\chi(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as $\frac{\Delta}{d+1} \to \infty$. The proof adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. We then extend that result to graphs where each neighbourhood has bounded density, which strengthens results due to Vu, and Davies, P., Kang and Sereni. As a final touch, we show that our method provides an asymptotically tight lower bound on the number of colourings of sparse graphs.