Number of distinguishing colorings and partitions

Symmetry breaking indices for the Cartesian product of graphs

“…) and for n ≥ 2 and k ≥ n we have Φ k (K n ) = k n [1]. They also provided the following theorem, in which the notation n k demonstrates the Stirling number of the second kind.…”

“…We remind the reader that the number of non-equivalent distinguishing colorings of a graph G with {1, • • • , k} as the set of admissible colors is shown by Φ k (G), while the number of nonequivalent k-distinguishing colorings of a graph G with {1, • • • , k} as the set of colors is shown by ϕ k (G). Ahmadi, Alinaghipour and Shekarriz defined these indices in [1], where they also pointed out that…”

“…Several relevant indices were recently introduced by Ahmadi, Alinaghipour and Shekarriz [1]. Two colorings c 1 and c 2 of a graph G are equivalent if there is an automorphism α of G such that c 1 (v) = c 2 (α(v)) for all v ∈ V (G).…”

“…When G has no distinguishing colorings with exactly k colors, we have ϕ k (G) = 0. For a graph G, the distinguishing threshold, θ(G), is the minimum number t such that for any k ≥ t, any arbitrary k-coloring of G is distinguishing [1].…”

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

“…To tackle these problems, different coloring schemes have been proposed: the scheme based on distances in [44], the scheme based on templates in [45], the scheme based on adjacencies in [46], the scheme based on heuristics in [47] and the scheme based on pseudo-randomness (with constrains, Grundy and color-dominating) in [48]. The properties of the colorings have been studied in [49] and the counting of distinguishing (symmetry breaking) colorings with k colors in [50]. One should notice that all Zagreb indices and their relatives [51] are useless for any topological isomers of fullerene, in which any vertex has a degree of 3 (in the related notation, d v = d w = 3).…”