Distinguishing threshold for some graph operations

Shekarriz, Mohammad Hadi; Fard, Seyed Alireza Talebpour Shirazi; Ahmadi, Bahman; Haghighi, Mohammad Hassan Shirdareh; Alikhani, Saeid

doi:10.48550/arxiv.2109.00045

Cited by 1 publication

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Shekarriz

Alikhani

2022

Art Discrete Appl. Math.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

“…where G k denotes the kth Cartesian power of G. Some other graph products were considered by Shekarriz et al [24] for their distinguishing thresholds. They studied θ(G) when G is the vertex-sum of some graphs or G is the smooth rooted product, the corona product or the lexicographic product of two graphs.…”

Section: Introductionmentioning

confidence: 99%

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

Shekarriz

Alikhani

2022

Art Discrete Appl. Math.

View full text Add to dashboard Cite

show abstract

Distinguishing threshold for some graph operations

Cited by 1 publication

References 0 publications

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

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

Contact Info

Product

Resources

About