On the locating chromatic number of some Buckminsterfullerene-type graphs

Let G = (V, E) be a nontrivial graph and k be a positive integer. Let c : V(G) → {1,2,3,…, k } be a vertex coloring of G such that if uv ∈ E(G) then c(u) ≠ c(v). For 1 ≤ i ≤ k , let S i be the i th set of vertices given color i and define Π = { S 1 , S 2 ,…, S k }. The color code of a vertex v ∈ V (G), denoted by c π ( v ), is defined as the ordered k-tuple c π (v) = (d(v, S 1 ), d(v, S 2 ), … , d(v, S k )), where d(v, S i ) = min{d(v, x) | x ∈ S i } for 1 ≤ i ≤ k . If every two vertices u and v in G have different color codes, then c is defined as the locating k -coloring of G . The minimum number of color used in the locating k -coloring of G is defined as the locating chromatic number of G , denoted by X L (G). This paper determined the locating chromatic number of some Buckminsterfullerene-type graph and some (4, 6)-fullerene graphs.