An Eulerian path visits all the edges of a graph in sequence, with no edges repeated. In 1736, Euler solved the Königsberg bridges problem by noting that the four regions of Königsberg each bordered an odd number of bridges, but that only two odd-valenced vertices could be in an Eulerian graph.
A semigraceful graph has edges labeled 1 to , with each edge label equal to the absolute difference between the labels of its vertices; the lowest vertex label is 0. For a fully graceful graph, the highest vertex label is .
This Demonstration looks at Eulerian semigraceful graphs. When the maximal vertex equals the number of edges, the graph is fully graceful.