Smaller Universal Targets for Homomorphisms of Edge-Colored Graphs

Abstract

For a graph G, the density of G, denoted D(G), is the maximum ratio of the number of edges to the number of vertices ranging over all subgraphs of G. For a class $\mathcal {F}$ of graphs, the value $D(\mathcal {F})$ is the supremum of densities of graphs in $\mathcal {F}$. A k-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots ,k$. 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 $\mathcal {F}$ of graphs, a k-edge-colored graph $\mathbb {H}$ (not necessarily with the underlying graph in $\mathcal {F}$) is k-universal for $\mathcal {F}$ when any k-edge-colored graph with the underlying graph in $\mathcal {F}$ admits a homomorphism to $\mathbb {H}$. Such graphs are known to exist exactly for classes $\mathcal {F}$ of graphs with acyclic chromatic number bounded by a constant. The minimum number of vertices in a k-uniform graph for a class $\mathcal {F}$ is known to be $\varOmega (k^{D(\mathcal {F})})$ and $O(k^{{\left\lceil D(\mathcal {F}) \right\rceil }})$. In this paper we close the gap by improving the upper bound to $O(k^{D(\mathcal {F})})$ for any rational $D(\mathcal {F})$.

Research was partially supported by MNiSW grant DI2013 000443.