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Title: Theorems and Exchange Graph Algorithms concerning Paths, Cycles and Trees

Abstract: Carsten Thomassen and I proved that in any graph G, the number of cycles containing a specified edge as well as all the odd-degree vertices is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Sunichi Toida and where all vertices have odd degree it is Andrew Thomason's generalization of Smith's Theorem.