On the local distinguishing numbers of cycles

Abstract

Consider the induced subgraph of a labeled graph G rooted at vertex v, denoted by N_vⁱ, where V(N_vⁱ) = {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, N_vⁱ is not isomorphic to N_uⁱ 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(C_n) is computed for each n. It is shown that $\text{LD}{}^{\mathrm{i}}\text{(C}{}_{\mathrm{n}}\text{) = Ω(n}{}^{\mathrm{<mtext>1</mtext><mtext>(2i+1)</mtext>}}\text{)}$. In addition, $\text{LD}{}^{\mathrm{i}}\text{(C}{}_{\mathrm{n}}\text{) ⩽ 24(2i + 1)n}{}^{\mathrm{<mtext>1</mtext><mtext>(2i+1)</mtext>}}\text{(}\text{log}\text{n)}{}^{\mathrm{<mtext>2i</mtext><mtext>(2i+1)</mtext>}}$ for constant i was proven using probabilistic methods. Finally, it is noted that for almost all graphs G, LD¹(G) = O(log n).