  Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the
largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor.
A hole in $G$ is an induced cycle of length at least four. 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 $h(G)\ge\chi(G)$. We also prove that if a graph $G$ with
independence number $\alpha(G)\ge2$ has no hole of length between $4$ and
$2\alpha(G)$, then $G$ contains an odd clique minor of size $\chi(G)$, that is,
such a graph $G$ satisfies the odd Hadwiger's conjecture.
