Number of colors needed to break symmetries of a graph by an arbitrary edge coloring

Abstract

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph G, denoted by θ(G), is the minimum number of colors k so that every k-coloring of G is distinguishing. We generalize this concept to edge-coloring by defining an alternative index θ ′ (G). We consider θ ′ for some families of graphs and find its connection with edge-cycles of the automorphism group. Then we show that θ ′ (G) = 2 if and only if G ≃ K 1,2 and θ ′ (G) = 3 if and only if G ≃ P 4 , K 1,3 or K 3 . Moreover, we prove some auxiliary results for graphs whose distinguishing threshold is 3 and show that although there are infinitely many such graphs, but they are not line graphs. Finally, we compute θ ′ (G) when G is the Cartesian product of simple prime graphs.