Locating-dominating codes in cycles

摘要

The smallest cardinality of an r-locating-dominating code in a cycle C_nof length n is denoted by m_r~(LD)(C_n).In this paper, we prove that forany r ≥ 5 and n≥n_rwhen n_ris large enough (a_r = O(r~3)) we haven/3≥m_r~(LD)(C_n)≥n/3 + 1 if n ≡3 (mod 6) and M_r~(LD)(C_n)= [n/3]otherwise. Moreover, we determine the exact values of M_3~(LD) (C_n)andM_4~(LD)(C_n) for all n.