A Note on the Asymptotics and Computational Complexity of Graph Distinguishability

A graph G is said to be d-distinguishable if there is a d-coloring of G which no non-trivial automorphism preserves. That is, 9χ : G !f1; : : : ;dg; 8φ 2 Aut(G)nfidg;9v;χ(v) 6= χ(φ(v)): It was conjectured that if jGj> jAut(G)j and the Aut(G) action on G has no singleton orbits, then G is 2-distinguishable. We give an example where this fails. We partially repair the conjecture by showing that when “enough motion occurs,” the distinguishing number does indeed decay. Specifically, defining m(G) = min φ2Aut(G) φ 6=id jfv 2G : φ(v) 6= vg j; we show that when m(G)> 2log2 jAut(G)j, G is indeed 2-distinguishable. In general, we show that if m(G) lnd > 2ln jAut(G)j then G is d-distinguishable. There has been considerable interest in the computational complexity of the ddistinguishability problem. Specifically, there has been much musing on the computational complexity of the language f(G;d) : G is d-distinguishableg : We show that this language lies in AM Σ2 \ΠP2 . We use this to conclude that if DIST is CONP-hard then the polynomial hierarchy collapses. AMS Classification: Primary: 05C25; Secondary: 68Q15. THE ELECTRONIC JOURNAL OF COMBINATORICS 5 (1998), #R23 2