Prove that in a connected graph G if $|G| < 2\delta(G)$ then the longest path in the graph consists of all vertices of the graph.

Question

I do know how to prove that the longest path should be at least $\delta(G)+1$ long. But I'm at a loss when I try to prove this. Any help is appreciated.

Answer 1

This is just Dirac theorem. Take a look at: https://en.wikipedia.org/wiki/Hamiltonian_path#Bondy%E2%80%93Chv%C3%A1tal_theorem

or here

http://faculty.wwu.edu/sarkara/dirac.pdf