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 \(\Omega (k^{D(\mathcal {F})})\) 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 \(D(\mathcal {F})\).
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Notes
In literature, universal graphs are also called homomorphism bounds.
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Acknowledgements
I would like to thank Grzegorz Gutowski and Prof. Paweł Idziak for their invaluable help with the preparation of this paper.
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The conference version of this paper appeared in COCOON 2019 (Guśpiel 2019).
Research was partially supported by MNiSW Grant DI2013 000443.
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Guśpiel, G. Smaller universal targets for homomorphisms of edge-colored graphs. J Comb Optim 43, 1534–1544 (2022). https://doi.org/10.1007/s10878-020-00680-3
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DOI: https://doi.org/10.1007/s10878-020-00680-3