\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
This paper proves that every $(n + 1)$-chromatic graph contains a
subgraph $H$ with $\chi_c (H) = n $. This provides easy methods
for constructing sparse graphs $G$ with $\chi_c (G) = \chi(G) = n
$. It is also proved that for any $\varepsilon > 0$, for any
fraction $k/d > 2$, there exsits an integer $g$ such that if $G$
has girth at least $g$ and $\chi_c (G) = k / d$ then for every
vertex $\upsilon$ of $G, \chi_c(G- \upsilon )
> k / d - \epsilon $. This implies that $G$ has an induced
subgraph $H$ with $\chi_c(G) - \epsilon < \chi_c (H) < \chi_c
(G)$.
\end{document}