Description:
Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph $G_{n,p}$ is concentrated in an interval of length at most $\omega\sqrt{n}$, and in the 1990s Alon showed that an interval of length $\omega\sqrt{n}/\log n$ suffices for constant edge-probabilities $p\in (0,1)$. We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case ${p=p(n) \to 0}$, and uncover a surprising concentration `jump' of the chromatic number in the very dense case ${p=p(n) \to 1}$.