Distinguishing threshold of graphs

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.

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The number of distinguishing colorings of a Cartesian product graph

Alikhani,

Shekarriz

2023

Quaestiones Mathematicae

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Distinguishing threshold of graphs

Abstract: A vertex coloring of a graph G is called distinguishing if no nonidentity automorphisms of G can preserve it. The distinguishing number of G, denoted by D G ( ), is the

Cited by 3 publications

References 30 publications

Distinguishing Threshold for Some Graph Operations

Distinguishing Threshold for Some Graph Operations

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

The number of distinguishing colorings of a Cartesian product graph

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