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Motivated by different characterizations of planar graphs and the 4-ColorTheorem, several structural results concerning graphs of high chromatic numberhave 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 asigned graph can be partitioned so that none of the parts induces a negativecycle. This extends the notion of the chromatic number of a graph since$\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graphobtained from~$G$ by replacing each edge with a pair of (parallel) positive andnegative edges. We introduce a signed version of Hadwiger's conjecture asfollows.  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'sconjecture and show its relation to the Odd Hadwiger Conjecture.  Motivated by these results, we also consider the relation betweensubdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ hasno negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced$\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result ofKawarabayashi (2013) on totally odd subdivisions.

Balanced-chromatic number and Hadwiger-like conjectures