  We study the mixing properties of the single-site Markov chain known as the
Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$
for constant $d$. The best known rapid mixing results for general graphs are in
terms of the maximum degree $\Delta$ of the input graph $G$ and hold when
$k>11\Delta/6$ for all $G$. Improved results hold when $k>\alpha\Delta$ for
graphs with girth $\geq 5$ and $\Delta$ sufficiently large where $\alpha\approx
1.7632\ldots$ is the root of $\alpha=\exp(1/\alpha)$; further improvements on
the constant $\alpha$ hold with stronger girth and maximum degree assumptions.
For sparse random graphs the maximum degree is a function of $n$ and the goal
is to obtain results in terms of the expected degree $d$. The following rapid
mixing results for $G(n,d/n)$ hold with high probability over the choice of the
random graph for sufficiently large constant~$d$. Mossel and Sly (2009) proved
rapid mixing for constant $k$, and Efthymiou (2014) improved this to $k$ linear
in~$d$. The condition was improved to $k>3d$ by Yin and Zhang (2016) using
non-MCMC methods. Here we prove rapid mixing when $k>\alpha d$ where
$\alpha\approx 1.7632\ldots$ is the same constant as above. Moreover we obtain
$O(n^{3})$ mixing time of the Glauber dynamics, while in previous rapid mixing
results the exponent was an increasing function in $d$. As in previous results
for random graphs our proof analyzes an appropriately defined block dynamics to
"hide" high-degree vertices. One new aspect in our improved approach is
utilizing so-called local uniformity properties for the analysis of block
dynamics. To analyze the "burn-in" phase we prove a concentration inequality
for the number of disagreements propagating in large blocks.
