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The optimal general upper bound for the distinguishing index of infinite graphs

The distinguishing index D′(G) of a graph G is the least cardinal number d such that G has an edge‐coloring with d colors, which is preserved only by the trivial automorphism. We prove a general upper bound D′(G)≤Δ−1 for any connected infinite graph G with finite maximum degree Δ≥3 . This is in contrast with finite graphs since for every Δ≥3 there exist infinitely many connected, finite graphs G with D′(G)=Δ . We also give examples showing that this bound is sharp for any maximum degree Δ .