An edge in a graph is an unordered pair of nodes. In a directed graph, the edges are ordered pairs.
One containing an Euler cycle. For (connected) undirected graphs, this is true iff every vertex has even degree. For directed graphs, this is true iff every vertex has equal indegree and outdegree. Planar bipartite graphs are dual to planar Eulerian graphs and vice versa.
A formula relating the number of edges, and nodes of a graph to the number of regions into which it will divide the plane. Let G be a connected planar graph with n nodes and e edges which divides the plane into r regions. Then any one of n, e, and r may be determined from the other two by the relation n-e+r=2.
A chain in a graph whose edges are all of the edges of the graph.
A cycle in a graph whose edges are all of the edges of the graph.