Another Antimagic Conjecture

An antimagic labeling of a graph G is bijection f:E(G)?{1,…,|E(G)|} such that the weights w(x)=?y?xf(y) distinguish all vertices. A well-known conjecture Hartsfield and Ringel (1990) every connected other than K2 admits an labeling. For set distances D, D-antimagic f:V(G)?{1,…,|V(G)|} weight?(x)=?y?ND(x)f(y) distinct for each vertex x, where ND(x)={y?V(G)|d(x,y)?D} D-neigbourhood x. If ND(x)=r, x in G, said to be (D,r)-regular. In this paper, we if only it does not contain two vertices having same D-neighborhood set. We also provide evidence true. present computational results that, D={1}, graphs order up 8 concur with conjecture. prove (D,r)-regular closed under union. examples disjoint union symmetric are non-symmetric non-(D,r)-regular D-antimagic. Furthermore, lastly, show possible obtain from previously known distance graph.