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Distinguishing number and adjacency properties
"... The distinguishing number of countably infinite graphs and relational structures satisfying a simple adjacency property is shown to be 2. This result generalizes both a result of Imrich et al. on the distinguishing number of the infinite random graph, and a result of Laflamme et al. on homogeneous ..."
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The distinguishing number of countably infinite graphs and relational structures satisfying a simple adjacency property is shown to be 2. This result generalizes both a result of Imrich et al. on the distinguishing number of the infinite random graph, and a result of Laflamme et al. on homogeneous relational structures whose age satisfies the free amalgamation property.
Orbit equivalence and permutation groups defined by unordered relations
 J. Algebr. Comb
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Finite Factors of Bernoulli Schemes and Distinguishing Labelings of Directed Graphs
"... A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the kblock presentation of a Berno ..."
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A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the kblock presentation of a Bernoulli scheme X. We determine the automorphism group of G, and thus the distinguishing labelings of G. A labeling of G defines a finite factor of X. We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of X. We use the Bell numbers to find a lower bound for the number of finitarily Markovian finite factors of a Bernoulli scheme. We show that demarcating labelings of G are distinguishing. 1
DISTINGUISHING HOMOMORPHISMS OF INFINITE GRAPHS
"... Abstract. We supply an upper bound on the distinguishing chromatic number of certain innite graphs satisfying an adjacency property. Distinguishing proper ncolourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph G satises the connected existentially c ..."
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Abstract. We supply an upper bound on the distinguishing chromatic number of certain innite graphs satisfying an adjacency property. Distinguishing proper ncolourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph G satises the connected existentially closed property and admits a homomorphism to H, then it admits continuummany distinguishing homomorphisms from G to H join K2: Applications are given to a family of universal Hcolourable graphs, for H a nite core. 1.