On (k, ℓ)-locating colorings of graphs

Let c: V(G) → {1,.., ℓ} = [ℓ] be a proper vertex coloring of G and C(i) = {u ∈ V(G): c(u) = i} for i ∈ [ℓ]. The k-color code r_k(v|c) of vertex v is the ordered ℓ-tuple (a_G(v,C(1)),.., a_G(v,C(ℓ))) where (Figure presented.) If every two vertices have different color codes, then c is a (k, ℓ)-locating coloring of G. The k-locating chromatic number of graph G, denoted by (Figure presented.), is the smallest integer ℓ such that G has a (k, ℓ)-locating coloring. In this paper, we propose this concept as an extension of diam(G)-locating chromatic number and 2-locating chromatic number which are known as the locating chromatic number, denoted χ_L(G), and neighbor-locating chromatic number, denoted (Figure presented.), respectively. In this paper, we give sharp bounds for (Figure presented.) and (Figure presented.) where G◦H and (Figure presented.) are the corona and edge corona of G and H, respectively. We formulate an integer linear programming model to determine (Figure presented.), noting that almost all graphs have diameter 2 and (Figure presented.) for every graph G of diameter 2.