On zig-zag chain graphs generated by regular hexagons with unbounded metric dimension

Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W = {w₁,w₂,...,w_k} is called a resolving set for G if for every two distinct vertices x,y ∈ V(G), there is a vertex w_i ∈ W such that d(x,w_i) ≠ d(y,w_i). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension denoted by dim(G). Let F be a family of connected graphs G_n: F = (G_n)_n≥1 depending on n as follows: the order | V(G) | = φ(n) and lim n→ ∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(G) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension; otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), then F is called a family with constant metric dimension. In this paper, we study the metric dimension of some zig-zag chain graphs generated by regular hexagons. We determine the exact value of their metric dimension and prove that these zig-zag chain graphs have unbounded metric dimension. It is an interesting and classical problem to classify the graphs families with respect to the nature of their metric dimension.