The dimension of a graph, that is the dimension of its incidence poset, became a major bridge between posets and graphs. Although allowing a nice characterization of planarity, this dimension badly behaves with respect to homomorphisms. We introduce the universal dimension of a graph G as the maximum dimension of a graph having a homomorphism to G. The universal dimension, which is clearly homomorphism monotone, is related to the existence of some balanced bicoloration of the vertices with respect to some realizer. Non trivial new results related to the original graph dimension are subsequently deduced from our study of universal dimension, including chromatic and extremal properties.