Eulerian path
Eulerian matroid, an abstract generalization of Eulerian graphs
Five room puzzle
Handshaking lemma, proven by Euler in his original paper, showing that any undirected connected graph has an even number of odd-degree vertices
Hamiltonian path – a path that visits each vertex exactly once.
Route inspection problem, search for the shortest path that visits all edges, possibly repeating edges if an Eulerian path does not exist.
Veblen's theorem, which states that graphs with even vertex degree can be partitioned into edge-disjoint cycles regardless of their connectivity
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