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Distinguishing graphs with infinite motion and nonlinear growth

2013; Society of Mathematicians, Physicists and Astronomers of Slovenia; Volume: 7; Issue: 1 Linguagem: Inglês

10.26493/1855-3974.334.fe4

1855-3966

Johannes Cuno, Wilfried Imrich, Florian Lehner,

graph theory and CDMA systems

The distinguishing number D( G ) of a graph G is the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs G with infinite motion and growth o ( n 2 / log 2 n ) is either 1 or 2 , which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs G of arbitrary cardinality are 2 -distinguishable if every nontrivial automorphism moves at least uncountably many vertices m( G ) , where m( G ) ≥ ∣Aut( G )∣ . This extends a result of Imrich et al. to graphs with automorphism groups of arbitrary cardinality.