Universal targets for homomorphisms of edge-colored graphs

TL;DR

This paper characterizes classes of graphs that admit universal edge-colored graphs for homomorphisms, establishing bounds on their size based on graph density and acyclic chromatic number, with implications for planar graphs.

Contribution

It provides a complete characterization of graph classes with universal graphs for homomorphisms and tight bounds on their size, linking acyclic chromatic number and density.

Findings

Classes with bounded acyclic chromatic number admit universal graphs.

Bounds on universal graph size are between $k^{D()}$ and $ck^{ceil{D()}}$.

For planar graphs, the size is bounded by a constant times $k^3$.

Abstract

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 $\mathbb{H}$. 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 $\mathcal{F}$ of graphs, $D(\mathcal{F})$ denotes the density of $\mathcal{F}$, that is the supremum of densities of graphs in $\mathcal{F}$. The main results are the following. The class $\mathcal{F}$ admits $k$-universal graphs for $k\geq2$ if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in $\mathcal{F}$. For any such class, there exists a constant $c$, such that for any $k \geq 2$, the size of the smallest $k$-universal graph is between $k^{D(\mathcal{F})}$ and $ck^{\lceil D(\mathcal{F})\rceil}$. A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (Journal of Algebraic Combinatorics, 8(1):5-13, 1998). One of their results is that for planar graphs, the size of the smallest $k$-universal graph is between $k^3+3$ and $5k^4$. Our results yield that there exists a constant $c$ such that for all $k$, this size is bounded from above by $ck^3$.