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.

翻译：受平面图的不同刻画及四色定理的驱动，关于高色数图的结构性结果已有诸多研究。为强化其中部分结果，我们考虑符号图 $\hat{G}$ 的*均衡色数* $\chi_b(\hat{G})$，即对符号图顶点进行划分所需的最小部数，使得每一部均不诱导负环。此概念推广了图的色数，因为 $\chi(G)=\chi_b(\tilde{G})$，其中 $\tilde{G}$ 表示将图 $G$ 的每条边替换为一对（平行）正负边后得到的符号图。我们提出如下哈德维格猜想的符号版本：猜想：若符号图 $\hat{G}$ 既无负环也无 $\tilde{K_t}$-子式，则其均衡色数至多为 $t-1$。我们证明该猜想事实上等价于哈德维格猜想，并揭示其与奇哈德维格猜想的关系。受这些结果启发，我们进一步考虑细分与均衡色数之间的关联。我们证明：若 $(G, \sigma)$ 既无负环也无 $\tilde{K_t}$-细分，则其允许一个均衡的 $\frac{79}{2}t^2$-染色。这定性推广了 Kawarabayashi（2013）关于全奇细分的结果。