a graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ e(g)} consists of n integers. inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of g and k2 , and the disjoint :union: of g...

Given a digraph D with m arcs and bijection τ:A(D)→{1,2,…,m}, we say (D,τ) is an antimagic orientation of graph G if no two vertices in have the same vertex-sum under τ, where vertex u τ sum labels all entering minus leaving u. Hefetz, Mütze, Schwartz 2010 initiated study orientations graphs, conjectured that every connected admits orientation. This conjecture seems hard, few related results ar...