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