  The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer
$t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to
Hadwiger in 1943 states that for every graph $G$, $h(G) \ge \chi(G)$, where
$\chi(G)$ denotes the chromatic number of $G$. Let $\alpha(G)$ denote the
independence number of $G$. A graph is $H$-free if it does not contain the
graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved
that $h(G) \ge \chi(G)$ for all $H$-free graphs $G$ with $\alpha(G) \le 2$,
where $H$ is any graph on four vertices with $\alpha(H) \le 2$, $H=C_5$, or $H$
is a particular graph on seven vertices. In 2010, Kriesell considered a
particular strengthening of Hadwiger's conjecture due to Seymour and
subsequently generalized the statement to include all forbidden subgraphs $H$
on five vertices with $\alpha(H) \le 2$. In this note, we prove that $h(G) \ge
\chi(G)$ for all $W_5$-free graphs $G$ with $\alpha(G) \le 2$, where $W_5$
denotes the wheel on six vertices.
