  We consider the problem of coloring a 3-colorable graph in polynomial time
using as few colors as possible. This is one of the most challenging problems
in graph algorithms.
  In this paper using Blum's notion of ``progress'', we develop a new
combinatorial algorithm for the following: Given any 3-colorable graph with
minimum degree $\ds>\sqrt n$, we can, in polynomial time, make progress towards
a $k$-coloring for some $k=\sqrt{n/\ds}\cdot n^{o(1)}$.
  We balance our main result with the best-known semi-definite(SDP) approach
which we use for degrees below $n^{0.605073}$. As a result, we show that
$\tO(n^{0.19747})$ colors suffice for coloring 3-colorable graphs. This
improves on the previous best bound of $\tO(n^{0.19996})$ by Kawarabayashi and
Thorup in 2017.
