If graph $G$ is Maximal Non-Hamiltonian. Does there exist a Hamiltonian Path?

Question

Can't find much on this question I want to know about. Would help me finish a problem if this was true.

Answer 1

Yes. Suppose that $G$ is maximally non-Hamiltonian. Add any edge $e$ to $G$ so that now you have a Hamilton cycle $C = v_1 \cdots v_n v_1$. If $C$ does not contain $e$, then $P = v_1 \cdots v_n$ is a Hamilton path in $G$. Else, up to permutation say that $e$ connects $v_n$ to $v_1$. Then $P = v_1 \cdots v_n$ is a Hamilton path in $G$, because it does not use $e$.

Note that this also show that if $G$ is such that by adding some edge $e$ to $G$ the resulting graph is Hamiltonian, then $G$ contains a Hamilton cycle.