Universal H-colorable Graphs Without A Given Configuration

For every pair of finite connected graphs F and H, and every integer k, we construct a universal graph U with the following properties: 1. There is a homomorphism : U ! H, but no homomorphism from F to U . 2. For every graph G with maximal degree no more than k having a homomorphism h : G ! H, but no homomorphism from F to G, there is a homomorphism ff : G ! U , such that h = ffi ff. Particularly, this solves a problem presented in [1] and [2] regarding the chromatic number of a universal graph. 1 Introduction For a graph G, let V (G) and E(G) denote its vertex and edge sets, respectively. Given two graphs G and H, a map f : V (G) ! V (H) is a (graph) homomorphism from G to H if and only if ff(x); f(y)g 2 E(H) for all edges fx; yg 2 E(G). For simplicity, f : G ! H will denote that f is a homomorphism from G to H, and G 6! H means that no homomorphism exists from G to H. Homomorphisms extend the notion of coloring, as (G) k if and only if G ! K k . More generally, we say that G..