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

If there are integers \( k \) and \( \lambda \neq 0 \) such that a total labeling \( f \) of a connected graph \( G = (V, E) \) from \( V \cup E \) to \( \{1, 2, \ldots, |V| + |E|\} \) satisfies \( f(x) \neq f(y) \) 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 \( (k, \lambda) \)-\emph{magically total labeling} (\( (k, \lambda) \)-\emph{mtl} for short) of \( G \). Several properties of \( (k, \lambda) \)-\emph{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 \( (k, \lambda) \)-\emph{mtls}.