On the partition dimension of circulant graph Cn(1, 2, 3, 4)

Abstract

Let Λ = {B1, B2, . . . , Bl} be an ordered l-partition of a connected graph G(V (G), E(G)). The partition representation of vertex x with respect to Λ is the l-vector, r(x|Λ) = (d(x, B1), d(x, B2), . . . , d(x, Bl)), where d(x, B) = min{d(x, y)|y ∈ B} is the distance between x and B. If the l - vectors r(x|Λ), for all x ∈ V (G) are distinct then l - partition is called a resolving partition. The least value of l for which there is a resolving l - partition is known as the partition dimension of G symbolized as pd(G). In this paper, the partition dimension of circulant graphs Cn(1, 2, 3, 4) is computed for n ≥ 8 as, pd(Cn(1, 2, 3, 4)) =    n, if 8 ≤ n ≤ 9; 6, if n = 10; 5, if n ≥ 11.