n-fold-L(d, 1)-labelings of the edge-multiplicity-path-replacements

An n-fold-L(d, 1)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u, v and any two integers a is an element of f (u), b is an element of f (v), vertical bar a-b vertical bar >= d if uv is an element of E(G), and vertical bar a-b vertical bar >= 1 if u and v are distance 2 apart. An n-fold-(d, 1)-total labeling of a graph G is an assignment f of sets of nonnegative integers of order n to each element of V (G) boolean OR E(G) of G such that: (i) any two adjacent vertices of G receive disjoint sets; (ii) any two adjacent edges of G receive disjoint sets; and (iii) a is an element of f (u), b is an element of f (e), vertical bar a-b vertical bar >= d for a vertex u and its incident edge e. This paper focuses on studying n-fold-L(d, 1) and n-fold-(d, 1)-total labeling of the edge-multiplicity-path-replacement G(rP(k)).