Symmetry breaking indices for the Cartesian product of graphs

Abstract

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing number of a graph is the minimum number of colors required for such a coloring. The distinguishing threshold of a graph G is the minimum number of colors k required that any arbitrary k-coloring of G is distinguishing. We prove a statement that gives a necessary and sufficient condition for a vertex coloring of the Cartesian product to be distinguishing. Then we use it to calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.

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

“…As noted in the introduction, using Lemma 2.4 the authors in [3] calculated the (vertex) distinguishing threshold for the Cartesian product graphs. However, they stated Lemma 2.4 so that it enables us to calculate edge-distinguishing threshold of the Cartesian products.…”

“…As an application, they used Φ k (G) to exactly calculate D(G•H), the distinguishing number of the lexicographic product of two connected graph G and H. Moreover, Shekarriz et al [24] used this index to calculate the distinguishing number of some graph operations such as the vertex-sum, the corona product and the smooth rooted product. Anyhow, the indices Φ k (G) and φ k (G) showed to be not easy ones to calculate in general, as to date explicit formulae are available only for paths, cycles, complete graphs and complete bipartaite graphs in [1] and just recently for grids (the Cartesian product of paths) by Alikhani and Shekarriz [3]. However, sometimes φ k (G) is very easy to calculate, eg.…”

“…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 □ . .…”