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A homomorphism of an oriented graph to an oriented graph is a mapping from to such that is an arc in whenever is an arc in . A homomorphism of to is said to be -preserving for some oriented graph if for every connected subgraph of isomorphic to a subgraph of , is isomorphic to its homomorphic image in . The -preserving oriented chromatic number of an oriented graph is the minimum number of vertices in an oriented graph such that there exists a -preserving...
We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these...
We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an -generated group is amenable if and only if the density of the corresponding Cayley graph equals to . We test amenable and non-amenable...
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