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The graphs used in this paper are simple, connected and finite graph. A graceful m -coloring of a graph G is a proper vertex coloring c : V (G) → { 1 , 2 , m } , where m ≥ 2 which induces a proper edge coloring c ′ : E (G) → { 1 , 2 , m − 1 } that defined by c ′ (u v) = | c (u) − c (v) |. Proper vertex coloring c of a graph G is a graceful coloring if c is a graceful m-coloring for m ∈ N. The minimum vertex coloring, that can apply such that a graph G can be colored with graceful coloring, is called a graceful chromatic number and it is denoted by χ g (G). In this paper, we will investigate the graceful chromatic number of generalized Petersen graph P (n , k) for k = 1 , 2 , 3 , 4. [ABSTRACT FROM AUTHOR]

Resolving independent dominating set is the development of metric dimension and independent dominating set. The set W is resolving independent dominating set for G if W is independent in G, and distinct vertices of G have distinct representations with respect to W. The minimum cardinality of resolving independent dominating set is called resolving independent domination number and denoted by γri(G). In this study, we examined the resolving independent dominating set of flower graphs, gear graphs, and sunflower graphs. [ABSTRACT FROM AUTHOR]

A simple graph G=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,...,V+E} is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,...,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,...,Gm, and then joining every vertex u in Gi, i∈{1,2,...,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles. [ABSTRACT FROM AUTHOR]