On the local distinguishing numbers of cycles Original Research Article

Consider the induced subgraph of a labeled graph G rooted at vertex v, denoted by Nvi, where V(Nvi) = {u: 0 ⩽ d(u,v) ⩽ i}. A labeling of the vertices of G, Φ : V(G) → {1,…, r} is said to be i-local distinguishing if ∀u, v ϵ V(G), u ≠ v, Nvi is not isomorphic to Nui under Φ. The ith local distinguishing number of G, LDi(G) is the minimum r such that G has an i-local distinguishing labeling that uses r colors. LDi(G) is a generalization of the distinguishing number D(G) as defined in Albertson and Collins (1996). An exact value for LD1(Cn) is computed for each n. It is shown that LDi(Cn) = Ω(n1(2i+1)). In addition, LDi(Cn) ⩽ 24(2i + 1)n1(2i+1)(log n)2i(2i+1) for constant i was proven using probabilistic methods. Finally, it is noted that for almost all graphs G, LD1(G) = O(log n).