Geodesic
Distinguishing infinite graphs
The electronic journal of combinatorics, Tome 14 (2007)
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The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products. For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$.
DOI : 10.37236/954
Classification : 05C25, 05C80, 03E10
Mots-clés : labeling, automorphism, countable random graph 
@article{10_37236_954,
     author = {Wilfried Imrich and Sandi Klav\v{z}ar and Vladimir Trofimov},
     title = {Distinguishing infinite graphs},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/954},
     zbl = {1124.05044},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/954/}
}
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Wilfried Imrich; Sandi Klavžar; Vladimir Trofimov. Distinguishing infinite graphs. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/954

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