On partition dimension of fullerene graphs

Abstract

Let G = (V(G); E(G)) be a connected graph and φ = fS 1; S2, …, Sk} be a kpartition of V(G). The representation r(vjφ) of a vertex v with respect to φ is the vector (Formula Presented). The partition φ is called a resolving partition of G if r(ujφ) ≠ r(vjφ) for all distinct u; v ∈ V(G). The partition dimension of G, denoted by pd(G), is the cardinality of a minimum resolving partition of G. In this paper, we calculate the partition dimension of two (4; 6)-fullerene graphs. We also give conjectures on the partition dimension of two (3; 6)-fullerene graphs.