On ( k , ℓ )-locating colorings of graphs

AbstractLet 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 rk(v|c) of vertex v is the ordered ℓ-tuple (aG(v,C(1)), . . . , aG(v,C(ℓ))) whereIf 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 , 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 χL(G), and neighbor-locating chromatic number, denoted , respectively. In this paper, we give sharp bounds for and where G◦H and are the corona and edge corona of G and H, respectively. We formulate an integer linear programming model to determine , noting that almost all graphs have diameter 2 and for every graph G of diameter 2.Mathematics Subject Classification (2020): 05C1505C69Key words: (k, ℓ)-locating coloringlocating coloringneighbor-locating coloringcorona productedge corona productILP model