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Abstract
Let $G = (V, E)$ be a graph. The \textit{Gallai total graph} $\Gamma_T(G)$ of $G$ is the graph, where $V(\Gamma_T(G))=V \cup E$ and $uv \in E(\Gamma_T(G))$ if and only if
\begin{itemize}
\item[$(i)$] $u$ and $v$ are adjacent vertices in $G$, or
\item[$(ii)$] $u$ is incident to $v$ or $v$ is incident to $u$ in $G$, or
\item[$(iii)$] $u$ and $v$ are adjacent edges in $G$ which do not span a triangle in $G$.
\end{itemize}
The \textit{anti-Gallai total graph} $\Delta_T(G)$ of $G$ is the graph, where $V(\Delta_T(G))=V \cup E$ and $uv \in E(\Delta_T(G))$ if and only if
\begin{itemize}
\item[$(i)$] $u$ and $v$ are adjacent vertices in $G$, or
\item[$(ii)$] $u$ is incident to $v$ or $v$ is incident to $u$ in $G$, or
\item[$(iii)$] $u$ and $v$ are adjacent edges in $G$ and lie on a same triangle in $G$.
\end{itemize}
In this paper, we discuss Eulerian and Hamiltonian properties of Gallai and anti-Gallai total graphs.
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