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arXiv · 2109.00045

Distinguishing threshold for some graph operations

Mohammad Hadi Shekarriz·Seyed Alireza Talebpour Shirazi Fard·Bahman Ahmadi·Mohammad Hassan Shirdareh Haghighi·Saeid Alikhani
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Abstract

A vertex coloring of a graph $G$ is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, $D(G)$, is the minimum number of colors required for such a coloring and the distinguishing threshold, $\theta(G)$, is the minimum number of colors~$k$ such that any arbitrary $k$-coloring is distinguishing. Moreover, $\Phi_k (G)$ is the number of distinguishing coloring of $G$ using at most $k$ colors. In this paper, for some graph operations, namely, vertex-sum, rooted product, corona product and lexicographic product, we find formulae of the distinguishing number and threshold using $\Phi_k (G)$.

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Mohammad Hadi Shekarriz, Seyed Alireza Talebpour Shirazi Fard, Bahman Ahmadi, Mohammad Hassan Shirdareh Haghighi, Saeid Alikhani. 2021-08-31. Distinguishing threshold for some graph operations. https://doi.org/10.1007/s40995-022-01379-2

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