A graph is [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) if its automorphism group acts transitively upon its vertices.  

A [Hamiltonian path](https://en.wikipedia.org/wiki/Hamiltonian_path) is a path that visits each vertex exactly once. 


[Lovász conjecture](https://en.wikipedia.org/wiki/Lov%C3%A1sz_conjecture): Every finite connected vertex-transitive graph contains a Hamiltonian path.  

This conjecture is open since almost 50 years. 


A [connected graph](https://en.wikipedia.org/wiki/Connectivity_(graph_theory)) is [$2$-connected](https://en.wikipedia.org/wiki/K-vertex-connected_graph) if it remains connected whenever any vertex is removed.  

**Question**: Is Lovász conjecture known in the $2$-connected case (or $k$-connected for $k$ large enough)?



Lovász conjecture and 2-connected graphs
The vertex connectivity of a vertex-transitive graph with valency $k$ is at least $2(k+1)/3$ (Mader/Watkins). So if you prove the conjecture for 3-connected graphs, you've done them all.


