An Euler path is a path along a connected graph that connects all the vertices and that traverses every edge of the graph only once. Note that a vertex with an odd degree allows one to travel through it and return by another path at least once, while a vertex with an even degree only allows a number of traversals through, but one cannot end an Euler path at a vertex with even degree. Thus, a connected graph has an Euler path which is a circuit (an Euler circuit) if all of its vertices have even degree. A connected graph has an Euler path which is non-circuituous if it has exactly two vertices with odd degree. Compare with Hamilton path.
Related category GRAPHS AND GRAPH THEORY
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