  Graph coloring problems are a central topic of study in the theory of
algorithms. We study the problem of partially coloring partially colorable
graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph
$G=(V,E)$ is $\alpha$-partially $k$-colorable, if there exists a subset
$S\subset V$ of cardinality $ |S | \geq \alpha | V |$ such that the graph
induced on $S$ is $k$-colorable. Partial $k$-colorability is a more robust
structural property of a graph than $k$-colorability. For graphs that arise in
practice, partial $k$-colorability might be a better notion to use than
$k$-colorability, since data arising in practice often contains various forms
of noise.
  We give a polynomial time algorithm that takes as input a $(1 -
\epsilon)$-partially $3$-colorable graph $G$ and a constant $\gamma \in
[\epsilon, 1/10]$, and colors a $(1 - \epsilon/\gamma)$ fraction of the
vertices using $\tilde{O}\left(n^{0.25 + O(\gamma^{1/2})} \right)$ colors. We
also study natural semi-random families of instances of partially $3$-colorable
graphs and partially $2$-colorable graphs, and give stronger bi-criteria
approximation guarantees for these family of instances.
