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Homomorphism and Dimension

The dimension of a graph, that is, the dimension of its incidence poset, has become a major bridge between posets and graphs. Although allowing a nice characterization of planarity, this dimension behaves badly 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 bicolouration of the vertices with respect to some realizer. Nontrivial new results related to the original graph dimension are subsequently deduced from our study of universal dimension, including chromatic properties, extremal properties and a disproof of two conjectures of Felsner and Trotter.