The IsEulerian command returns true if the input graph is an Eulerian graph, i.e there exists a closed walk in the graph that uses each edge exactly once. It returns false otherwise.

An optional second argument T is assigned an Eulerian Trail of the graph if such a trail exists, and FAIL otherwise.

The FindEulerianPath command returns a list corresponding to an Eulerian trail if one exists, and NULL otherwise.

The algorithm used to construct the Eulerian trail is depth-first-search. The complexity is $\mathrm{O}\left(n+m\right)$ where n=|V| and m=|E|.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{IsEulerian}\left(\mathrm{CompleteGraph}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsEulerian}\left(\mathrm{CompleteGraph}\left(5\right)\,'T'\right)$

$\mathrm{true}$

$T$

$\mathrm{Trail}\left(1\,2\,3\,1\,4\,2\,5\,3\,4\,5\,1\right)$

$\mathrm{FindEulerianPath}\left(\mathrm{SpecialGraphs}:-\mathrm{BookGraph}\left(3\right)\right)$

$\left[1\,2\,4\,3\,1\,5\,6\,2\,8\,7\,1\right]$

The GraphTheory[FindEulerianPath] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.