Is every Eulerian triangulated (planar) graph Hamiltonian?
On the other hand we have that:
Barnette's Conjecture (Open): Every cubic bipartite (3-connected) planar graph is Hamiltonian.
Notice that: Eulerian triangulated planar graphs and cubic bipartite planar graphs are mutual duals.
I would be happy to know of a proof or a counterexample (I have not been able to find either). In the absence of both is this a well known conjecture?
Some additional facts (possibly relevant):
- Every Eulerian triangulated planar graph is $3$-colorable (see this).
- (Whitney's Theorem) Every $4$-connected triangulated planar graph is Hamiltonian.
- Hamiltonian Graph: Undirected Graph with a simple cycle through every vertex.
- Eulerian Graph: Connected graph with all degrees even.
- Triangulated (planar) Graph: Planar graph with every face a triangle.
- Cubic Graph: Graph with every vertex having degree exactly $3$.
Edit: (Thanks@domotorp) Triangulated planar graphs are not necessarily Hamiltonian (e.g. see this)