\documentclass[12pt]{article}


\usepackage{graphicx} %The use of this package is essential for the inclusion of graphics.
\usepackage{doc} %To interpret the command \BibTex.

\title{Some \LaTeX\ facilities}
\author{J. Lauri\\ \it Department of Mathematics\\
\it University of Malta\\ \it Malta}
\date{}

\begin{document}

\maketitle

\begin{abstract}
This short document is intended to show those students needing to
use \LaTeX\ a simple example of how to include EPS files and how
to cite references without using \BibTeX. The original \LaTeX\
file is also available.
\end{abstract}

\section{Introduction}

Let $G$ be a connected {\em 2-in-2-out digraph}, that is, a
connected digraph in which each vertex has in-degree and
out-degree both equal to 2 (loops and multiple arcs are allowed).
Such a digraph is Eulerian. Let $Eu(G)$ be the set of Euler trails
in $G$ and let $\gamma\in Eu(G)$.

\bigskip\noindent
In \cite{mp} the following theorem is proved. \emph{You have just
seen an example of citation of a reference.}


In this note we shall give an elementary combinatorial proof of
this result.

\begin{figure}[h]
\centering
\includegraphics[width=8cm, height=7cm]{fig_for_example1.eps}
\caption{This is the caption for the figure. Change the width and
height for a better result}
\end{figure}



\section{Decomposition of cycles by transpositions}

A matrix very similar to $I_\gamma$ as defined above has already
been considered by others \cite{cl,b}. \emph{This is another
example of a citation to a reference.}

\bigskip\noindent
For our purposes, the main result from \cite{cl,b} is the
following. \emph{In this article, the citations are incorporated
within the \LaTeX\ file (see the end of the file). This is
sufficient for a small list like this. For a longer list of
references, use \BibTeX.}


Cohn and Lempel \cite{cl} proved this result when the transpositions
are disjoint, and Beck \cite{cl} generalised it for arbitrary transpositions.
The case of disjoint transpositions will be sufficient for
our purposes.


\section{Proof of Theorem 1.1}

The proof of Theorem 1.1 will follow as a result of the
following.

%The list of references follows. With BibTeX you can let the programme do
%the formatting and the alphabetical listing.


\begin{thebibliography}{1}

\bibitem{b}
I.~Beck.
\newblock Cycle decomposition by transpositions.
\newblock {\em J. Combin. Theory (A)}, 23:198--207, 1977.

\bibitem{cl}
M.~Cohn and A.~Lempel.
\newblock Cycle decomposition by disjoint transpositions.
\newblock {\em J. Combin. Theory (A)}, 13:83--89, 1972.

\bibitem{mp}
N.~Macris and J.V. Pul\'e.
\newblock Note: An alternative formula for the number of Euler trails for a
  class of digraphs.
\newblock {\em Discrete Mathematics}, To appear.

\end{thebibliography}



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