On partition dimension of fullerene graphs

Let G = (V(G), E(G)) be a connected graph and Pi = {S-1, S-2, ... ,S-k} be a k-partition of V(G). The representation r(v vertical bar Pi) of a vertex v with respect to Pi is the vector (d(v, S-1), d(v, S-2), ... ,d(v, S-k)), where d(v, S-i) = min{d(v, s(i))vertical bar s(i) is an element of S-i). The partition H is called a resolving partition of G if r(u vertical bar Pi) not equal r(v vertical bar Pi) for all distinct u, v is an element of 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.