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Let $G$ be a graph with a nonempty edge set, we denote the rank of
the adjacency matrix of $G$ and term rank of $G$, by rk$(G)$ and
Rk$(G)$, respectively. C. Van Nuffelen conjectured that for any
graph $G$, $\chi(G)\leq rk$$(G)$. The first counter example to
this conjecture was obtained by Alon and Seymour. In 2002,
Fishkind and Kotlov proved that for any graph $G$, $\chi(G)\leq
Rk$$(G)$. Here we improve this upper bound and show that
$\chi_1(G)\leq\frac{rk(G)+Rk(G)}{2}$ where $\chi_1(G)$ is the list
chromatic number of $G$.
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