【正文】 to and from them by arcs. ? Usually the graph is drawn like this (an isomorphic graph.) The problem now bees one of drawing this picture without retracing any line and without picking your pencil up off the paper. Euler saw that there were 5 vertices that each had an odd number of lines connected to it. He stated they would either be the beginning or end of his pencilpath. Paths and Circuits Euler path a continuous path that passes through every edge once and only once. Euler circuit when a Euler path begins and ends at the same vertex If a graph has any vertices of odd degree, then it can39。t have any Euler circuit. If a graph is connected and every vertex has an even degree, then it has at least one Euler circuit (usually more). Euler’s 1st Theorem Proof: S’pose we have an Euler circuit. ? If a node has an odd degree, and the