Brélaz is a heuristic for vertex coloring from the 70s, also known as DSATUR (degree of saturation). It is known to be specially good with Erdös-Rényi graphs (uniform random) and achieves a good compromise between effort and tightness.
However it is only just that, a heuristic, and cannot certify an optimum.
Brélaz can be also used inside a full enumeration scheme, as described in [Randall-Brown 72], but I guess this is not what you have implemented.
There have also been some improvements over DSATUR, mainly concerning tiebreaks
- Sewell 1996
- San Segundo 2012 (algorithm PASS)