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Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\\ge \\chi(G)$, where $\\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $\\alpha(G)\\ge3$ has no hole of length between $4$ and $2\\alpha(G)-1$, then $h(G)\\ge\\chi(G)$. 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