Can a bipartite graph have many Hamiltonian paths but no Hamiltonian cycle? Can a bipartite graph with at least three vertices have the following properties simultaneously:


*

*Every vertex is the initial vertex of some Hamiltonian path.

*The graph contains no Hamiltonian cycle.


I found out that this is only possible if the number of vertices is even and if the partitions have equal size. My conjecture is that there is no bipartite graph with the desired properties, but I cannot prove it.

a bipartite graph is acyclic, all vertices of two sets must have at least one connection, graph is two colorable, has distinct edge
