Universal targets for homomorphisms of edge-colored graphs

Abstract

A k-edge-colored graph is a finite, simple graph with edges labeled by numbers 1,…,k. A function from the vertex set of one k-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class F of graphs, a k-edge-colored graph H (not necessarily with the underlying graph in F) is k-universal for F when any k-edge-colored graph with the underlying graph in F admits a homomorphism to H. We characterize graph classes that admit k-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph.For a nonempty graph G, the density of G is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of G. For a nonempty class F of graphs, D(F) denotes the density of F, that is the supremum of densities of graphs in F.The main results are the following. The class F admits k-universal graphs for k⩾2 if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in F. For any such class, there exists a constant c, such that for any k⩾2, the size of the smallest k-universal graph is between kD(F) and ck⌈D(F)⌉.A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (1998) [2]. One of their results is that for the class of planar graphs, the size of the smallest k-universal graph is between k3+3 and 5k4. Our results yield that there exists a constant c such that for all k, this size is bounded from above by ck3.