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Let $G$ be a graph with nonempty edge set, we denote the rank of
the adjacency matrix of $G$, by rk$(G)$ and Rk$(G)$, respectively.
It was conjectured [4], for any graph $G,
\chi(G)\leq\mathrm{rk}(G)$. The first counterexample to this
conjecture was obtained by Alon and Seymour [1]. Recently,
Fishkind and Kotlov [3] have proved that for any graph $G,
\chi(G)\leq\mathrm{Rk}(G)$. In this paper, we improve
Fishkind-Kotlov upper bound and show that
$\chi(G)\leq\frac{\mathrm{rk}(G)+\mathrm{Rk}(G)}{2}$.
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