Edge Antimagic Total Labeling on Paths and Unicycles

Abstract

Let $G=(V,E)$ be a simple and undirected graph with $v$ vertices and $e$ edges. An $(a,d)$-\emph{edge-antimagic total labeling} is a bijection $f$ from $V(G)\cup E(G)$ to the set of consecutive integers $\{1,2,\dots ,v+e\}$ such that the weights of the edges form an arithmetic progression with initial term $a$ and common difference $d$. A super $(a,d)$-\emph{edge antimagic total labeling} is an edge antimagic total labeling $f$ such that $f(V(G))=\{1,\dots ,v\}$. In this paper, we solve some problems on edge antimagic total labeling, such as on paths and unicyclic graphs.