On the partition dimension of a class of circulant graphs

Abstract

For a vertex v of a connected graph G(V,E) and a subset S of V, the distance between a vertex v and S is defined by d(v,S)=min{d(v,x):x∈S}. For an ordered k-partition π={S1,S2…Sk} of V, the partition representation of v with respect to π is the k-vector r(v|π)=(d(v,S1),d(v,S2)…d(v,Sk)). The k-partition π is a resolving partition if the k-vectors r(v|π), v∈V(G) are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al. [1] in their paper which appeared in Acta Mathematica Sinica, English Series proved that partition dimension of a class of circulant graph G(n,±{1,2}), for all even n⩾6 is four. In this paper we prove that it is three.