  in graph theory, a eulerian trail (or eulerian path) is a trail in a graph which visits every edge exactly once. similarly, an eulerian circuit or eulerian cycle is an eulerian trail which starts and ends on the same vertex. they were first discussed by leonhard euler while solving the famous seven bridges of kÃ¶nigsberg problem in 1736. mathematically the problem can be stated like this: :given the graph in the image, is it possible to construct a path (or a cycle, i.e. a path starting and ending on the same vertex) which visits each edge exactly once? euler proved that a necessary condition for the existence of eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an eulerian circuit. the first complete proof of this latter claim was published posthumously in 1873 by carl hierholzer.n. l. biggs, e. k. lloyd and r. j. wilson, graph theory 1736â1936, clarendon press, oxford, 1976, 8â9, isbn 0-19-853901-0. the term eulerian graph has two common meanings in graph theory. one meaning is a graph with an eulerian circuit, and the other is a graph with every vertex of even degree. these definitions coincide for connected graphs. for the existence of eulerian trails it is necessary that zero or two vertices have an odd degree; this means the kÃ¶nigsberg graph is not eulerian. if there are no vertices of odd degree, all eulerian trails are circuits. if there are exactly two vertices of odd degree, all eulerian trails start at one of them and end at the other. a graph that has an eulerian trail but not an eulerian circuit is called semi-eulerian. ==definition== an eulerian trail,some people reserve the terms path and cycle to mean non-self-intersecting path and cycle. a (potentially) self-intersecting path is known as a trail or an open walk; and a (potentially) self-intersecting cycle, a circuit or a closed walk. this ambiguity can be avoided by using the terms eulerian trail and eulerian circuit when self-intersection is allowed. or euler walk in an undirected graph
Graph_theory_objects