How colorful the signed graph?

The zero-free chromatic number χ^* of a signed graph ∑ is the smallest positive number k for which the vertices can be colored using ±1, ±2,...,±k so that endpoints of a positive edge are not colored the same and those of a negative edge are not colored oppositely. We establish the value of χ^* for some special signed graphs and prove in general that χ^* equals the minimum size of a vertex partition inducing an antibalanced subgraph of ∑, and also the minimum chromatic number of the positive subgraph of any signed graph switching equivalent to ∑. We characterize those signed graphs with the largest and smallest possible χ^*, that is n, n-1, and 1, and the simple ones with the maximum and minimum χ^*, that is [ n 2] and 1, where n is the number of vertices. We give tighter bounds on χ^* in terms of the underlying graphs, but they are not sharp. We conclude by observing that determining χ^* is an NP-complete problem.