  Motivated by different characterizations of planar graphs and the 4-Color
Theorem, several structural results concerning graphs of high chromatic number
have been obtained. Toward strengthening some of these results, we consider the
\emph{balanced chromatic number}, $\chi_b(\hat{G})$, of a signed graph
$\hat{G}$. This is the minimum number of parts into which the vertices of a
signed graph can be partitioned so that none of the parts induces a negative
cycle. This extends the notion of the chromatic number of a graph since
$\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graph
obtained from~$G$ by replacing each edge with a pair of (parallel) positive and
negative edges. We introduce a signed version of Hadwiger's conjecture as
follows.
  Conjecture: If a signed graph $\hat{G}$ has no negative loop and no
$\tilde{K_t}$-minor, then its balanced chromatic number is at most $t-1$.
  We prove that this conjecture is, in fact, equivalent to Hadwiger's
conjecture and show its relation to the Odd Hadwiger Conjecture.
  Motivated by these results, we also consider the relation between
subdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ has
no negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced
$\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of
Kawarabayashi (2013) on totally odd subdivisions.
