On $(k,\lambda )$-Magically Total Labeling of Graphs

Abstract

If there are integers $k$ and $\lambda \ne 0$ such that a total labeling $f$ of a connected graph $G=(V,E)$ from $V\cup E$ to $\{1,2,\dots ,|V|+|E|\}$ satisfies $f(x)\ne f(y)$ for distinct $x,y\in V\cup E$ and

$$f(u)+f(v)=k+\lambda f(uv)$$

for each edge $uv\in E$, then $f$ is called a $(k,\lambda )$-$magicallytotallabeling$ ($(k,\lambda )$-$mtl$ for short) of $G$. Several properties of $(k,\lambda )$-$mtls$ of graphs are shown. The sufficient and necessary connections between $(k,\lambda )$-\emph{mtls} and several known labelings (such as graceful, odd-graceful, felicitous, and $(b,d)$-edge antimagic total labelings) are given. Furthermore, every tree is proven to be a subgraph of a tree having super $(k,\lambda )$-$mtls$.