Is there a specific name for this necessary condition of Hamiltonian graphs? My professor told me that almost every graph theory textbook lists the following as a necessary condition for a graph to be Hamiltonian:
Let G be a graph. Every subset S of vertices of G has the property that the graph G - S has at most |S| components.
Is there a common name for this necessary condition? Part of my research relies on it and having access to other material about it would be helpful. Thank you for your time and have a great day! :)
P.S. I can give an example of the condition if the wording is confusing.
 A: This condition was introduced by Chvátal in the paper Tough graphs and hamiltonian circuits.
Here is some terminology that can help you along. We say that a graph $G$ is $t$-tough if for every $S \subseteq V(G)$, $G-S$ is either connected or else has at most $|S|/t$ components. A $1$-tough graph can simply be called "tough". Finally, the toughness of a graph $G$ is the maximum value of $t$ for which $G$ is $t$-tough.
The condition in your question can now be stated as

All Hamiltonian graphs are tough

and if I had to give it a name, I would call it the "toughness condition" or maybe "Chvátal's toughness condition".
Of course, the converse is not true; not all tough graphs are Hamiltonian. Chvátal constructs a $7$-vertex counterexample in the paper I linked to, but it may be easier to observe that Petersen's graph is a tough graph which is not Hamiltonian. (Checking this is a bit tedious.)
Chvátal's $t_0$-tough conjecture is that there is some $t_0> 1$ such that every $t_0$-tough graph is Hamiltonian; this is still open. Chvátal also conjectured more strongly that every $2$-tough graph is Hamiltonian, but counterexamples to that have since been found.
