Euler circuit EulerCircuit 2001-11-28 15:09:10 2004-04-22 08:33:53 Definition Euler path Euler circuit \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} An Euler circuit is a connected graph such that starting at a vertex $a$, one can traverse along every edge of the graph once to each of the other vertices and return to vertex $a$. In other words, an Euler circuit is an Euler path that is a circuit. Thus, using the properties of odd and even \PMlinkid{degree}{788} vertices given in the definition of an Euler path, an Euler circuit exists if and only if every vertex of the graph has an even degree. \begin{center} \includegraphics[width=1in,height=1in]{ecircuit} \end{center} This graph is an Euler circuit as all vertices have degree 2. \begin{center} \includegraphics[width=1in,height=1in]{epath} \end{center} This graph is not an Euler circuit.