Super edge-antimagic graceful labeling of graphs

Abstract

For a graph $G=(V, E)$, a bijection $g$ from $V(G) \cup E(G)$ into $\{1,2, \ldots,|V(G)|+|E(G)|\}$ is called $(a, d)$-edge-antimagic graceful labeling of $G$ if the edge-weights $w(x y)=|g(x)+g(y)-g(x y)|, x y \in E(G)$, form an arithmetic progression starting from $a$ and having a common difference $d$. An $(a, d)$-edge-antimagic graceful labeling is called super $(a, d)$-edge-antimagic graceful if $g(V(G))=\{1,2, \ldots,|V(G)|\}$. Note that the notion of super $(a, d)$-edge-antimagic graceful graphs is a generalization of the article "G. Marimuthu and $\mathrm{M}$. Balakrishnan, Super edge magic graceful graphs, Inf.Sci.287( 2014)140-151", since super $(a, 0)$-edge-antimagic graceful graph is a super edge magic graceful graph.We study super $(a, d)$-edge-antimagic graceful properties of certain classes of graphs, including complete graphs and complete bipartite graphs.