Distinguishing threshold of graphs

Abstract

A vertex coloring of a graph G is called distinguishing if no non-identity automorphisms of G can preserve it. The distinguishing number of G, denoted by D(G), is the minimum number of colors required for such coloring. The distinguishing threshold of G, denoted by θ(G), is the minimum number k of colors such that every k-coloring of G is distinguishing. In this paper, we study θ(G), find its relation to the cycle structure of the automorphism group and prove that θ(G) = 2 if and only if G is isomorphic to K 2 or K 2 . Moreover, we study graphs that have the distinguishing threshold equal to 3 or more and prove that θ(G) = D(G) if and only if G is asymmetric, K n or K n . Finally, we consider Johnson scheme graphs for their distinguishing number and threshold concludes the paper.

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

“…. , α r−1 (v)) is a cycle of length r if r is the least integer such that α r (v) = v. The number of cycles of an automorphism α is shown by |α| [23]. The distinguishing threshold is highly connected with this number.…”

“…The distinguishing threshold is highly connected with this number. Lemma 2.1 ( [23]). For any graph G, we have…”

“…Using this lemma, one can show that θ(G) = 2 if and only if |G| = 2 and θ(G) = 3 only if |G| = 3 or G is a graph with some certain conditions of order 2p for a prime number p. More explicitly, since we need it in Section 3, we have the following result. Let |G| = n. Then θ(G) = D(G) if and only if G is either asymmetric, the complete graph K n or the empty graph K n [23].…”

“…for n ≥ 3, and for n ≥ 5, they have proved that θ(K(n, 2)) = 1 2 (n 2 − 3n + 6) where K(n, k) is the Kneser graph [1]. Shekarriz et al [23] continued studying the distinguishing threshold by showing that θ(G) is related to the cycle structure of the automorphism group of G and calculating the distinguishing threshold for all graphs in the Johnson scheme. The distinguishing threshold for the Cartesian product of connected graphs has been studied by Alikhani and Shekarriz [3], who showed that when G = G 1 □G 2 □ .…”

“…Using this action, every automorphism decomposes into cycles of edges. Similar to Shekarriz et al [23], we represent an automorphism of G by a product of cyclic permutations of edges where this representation is unique up to a permutation of cycles. So, the ordered tuple σ = (e, α(e), α 2 (e), .…”