First, I am an undergraduate so I apologize if this is trivial and certainly understand if it is closed immediately.

I am currently in a combinatorics and graph theory class and recently we have been studying Hamiltonian graphs. We have been discussing a few theorems characterizing these graphs. I am interested in Dirac's theorem which states

Dirac (1952) Let $G$ be a simple graph with $n \geq 3$ vertices such that for any vertex $v \in G$ we have $\deg(v) \geq \frac{n}{2}$. Then $G$ is Hamiltonian.

The converse is easily seen to be false. I am interested in understanding how often the converse fails. From my view, one way to make this precise is as follows. Let $H_n$ denote the set of Hamiltonian graphs on $n$-vertices. What can we say about the probability $$p_n = P(\deg(v) \geq \frac{n}{2},\forall v\in G \mid G \in H_n)$$

I am mainly interested in what happens as $n \to \infty$. For example, I think it might be interesting if Dirac's theorem becomes necessary and sufficient if we take $n$ large enough. One could also investigate analogous question for other theorems that give sufficient conditions for $G$ to be Hamiltonian (Ore's theorem, Posa's theorem). However, Dirac's seemed the simplest to investigate.

Is there any literature on questions resembling this?

Thanks.

Edit:

I have fixed my maple code and believe I now have the correct probabilities for $n=3,4,5,6,7,8,9$ computed:

• $p_3 = 1$
• $p_4 = 1$
• $p_5 = \frac{3}{8} = 0.375$
• $p_6 = \frac{19}{48} \approx 0.396$
• $p_7 = \frac{29}{383} \approx 0.075$
• $p_8 = \frac{106}{1549} \approx 0.068$
• $p_9 = \frac{1165}{177083} \approx 0.007$

So maybe $p_n \to 0$ as $n \to \infty$? Of course this is clearly not enough data to make any reasonable conjectures. Much more powerful computers than mine would likely be needed. Thanks everyone for the useful comments, I imagine this question could probably be closed.