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

An Eulerian path is a walk that traverses each edge exactly once, but whose initial and final vertices are not required to be the same. Every Eulerian cycle is an Eulerian path, but the reverse is not true.

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

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

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

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

The following are graphs in the SpecialGraphs subpackage which are Eulerian.

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

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

$\mathrm{false}$

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

$\mathrm{true}$

$T$

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

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

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

The GraphTheory[FindEulerianCycle] and GraphTheory[FindEulerianPath] commands were introduced in Maple 2020.

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