Universal H-colorable graphs without a given configuration

AbstractFor every pair of finite connected graphs F and H, and every positive integer k, we construct a universal graph U with the following properties:(1)There is a homomorphism π:U→H, but no homomorphism from F to U.(2)For every graph G with maximum degree no more than k having a homomorphism h:G→H, but no homomorphism from F to G, there is a homomorphism α:G→U, such that h=π∘α.Particularly, this solves a problem presented in [1] and [2] regarding the chromatic number of a universal graph