Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture
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A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.
2010 ◽
Vol 370
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pp. 146-158
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1993 ◽
Vol 45
(4)
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pp. 863-878
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2013 ◽
Vol 71
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pp. 22-29
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2016 ◽
Vol 25
(06)
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pp. 1650033
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2011 ◽
Vol 226
(3)
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pp. 2643-2675
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