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labeled graph

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Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$.  A \emph{labeling} of $G$ is a partial function $\ell: V\cup E\to L$ for some set $L$.  For every $x$ in the domain of $\ell$, the element $\ell(x)\in L$ is called the \emph{label} of $x$.  Three of the most common types of labelings of a graph $G$ are
\begin{itemize}
\item \emph{total labeling}: $\ell$ is a total function (defined for all of $V\cup E$),
\item \emph{vertex labeling}: the domain of $\ell$ is $V$, and
\item \emph{edge labeling}: the domain of $\ell$ is $E$.
\end{itemize}
Usually, $L$ above is assumed to be a set of integers.  A \emph{labeled graph} is a pair $(G,\ell)$ where $G$ is a graph and $\ell$ is a labeling of $G$.

An example of a labeling of a graph is a coloring of a graph.  Uses of graph labeling outside of combinatorics can be found in areas such as order theory, language theory, and proof theory.  A proof tree, for instance, is really a \emph{labeled tree}, where the labels of vertices are formulas, and the labels of edges are rules of inference.

\textbf{Remarks}.  
\begin{itemize}
\item
Every labeling of a graph can be extended to a total labeling.
\item
The notion of labeling can be easily extended to digraphs, multigraphs, and pseudographs.
\end{itemize}
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