IsEulerian(G) returns true if G is an Eulerian graph; that is, if G has an Eulerian cycle. It returns false otherwise.

IsSemiEulerian(G) returns true if G is a semi-Eulerian graph; that is, if G has an Eulerian path. It returns false otherwise.

An optional second argument T is assigned a Trail corresponding to a walk of the specified type if it exists, and FAIL otherwise.

FindEulerianCycle(G) command returns a list corresponding to an Eulerian cycle in G if one exists, and NULL otherwise.

FindEulerianPath(G) command returns a list corresponding to an Eulerian path in G 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|.

An Eulerian cycle is a closed walk in the graph that traverses each edge exactly once. In this context "closed" means the walk must start and finish on the same vertex.

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.

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

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

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

$\mathrm{false}$

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

$\mathrm{true}$

$T$

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

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

$\mathrm{BG}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 8\; vertices\; and\; 10\; edges}$

$\mathrm{IsEulerian}\left(\mathrm{BG}\right)$

$\mathrm{true}$

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

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

$\mathrm{PG}≔\mathrm{PathGraph}\left(5\right)$

$\mathrm{PG}≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 4\; edges}$

$\mathrm{IsEulerian}\left(\mathrm{PG}\right)$

$\mathrm{false}$

$\mathrm{IsSemiEulerian}\left(\mathrm{PG}\right)$

$\mathrm{true}$

$\mathrm{FindEulerianPath}\left(\mathrm{PG}\right)$

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

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

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

The GraphTheory[IsSemiEulerian] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.