My case is a directed graph with $n$ nodes with $(n-1)^2+1$ edges. I have done the following till now.

We know that the maximum number of edges for a directed graph $K_n$ on $n$ nodes is $n(n-1)$ edges. The graph in my problem statement is $G(V,E)$ with $|V| = n$ and $|E|$ = $(n-1)^2+1$.

Now, $n(n-1) - ((n-1)^2 + 1) = n-2$, so any such graph can be obtained from $K_n$ by deleting exactly $n-2$ edges from $K_n$.

Is my approach correct till now? How can I apply induction to prove the graph is Hamiltonian? I'm new to graph theory and inductions. As such, a comprehensive simple explanation would be much appreciated.

If not induction, is there any other way to prove this?

  • $\begingroup$ There are some inconsistencies in your post, like confusing edges and nodes, and uneeded repetitions. Could you please correct them? $\endgroup$ – Nathaniel Jun 6 at 9:11
  • $\begingroup$ I highly recommend reading Jeff Erickson's notes on induction. Yes, it's 30 pages, but induction is a key skill and I haven't seen a friendlier introduction. $\endgroup$ – j_random_hacker Jun 6 at 10:16
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    $\begingroup$ This site is not for time-sensitive questions. $\endgroup$ – Yuval Filmus Jun 6 at 15:33
  • $\begingroup$ I do not understand this bound. If I omit all $n-1$ outgoing edges from one of the nodes of $K_n$ then the graph no longer is Hamiltonian (if we expect a cycle in that definition). $\endgroup$ – Hendrik Jan Jun 7 at 14:23
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    $\begingroup$ @HendrikJan So, at most $n-2$ edges can be omitted. That is the bound given in the question. $\endgroup$ – John L. Jun 8 at 0:23

The complete digraph of $n$ nodes, $K_n$ has $n(n-1)$ edges. Describe a digraph of $n$ nodes with $n(n-1)-\delta$ edges as a digraph "with $\delta$ edges removed".

A proof by induction

The following is an outline to prove by induction that every digraph of $n$ nodes with $n-2$ edges removed contains a Hamiltonian cycle.

The base case, when $n=2$ or $n=3$ is obviously correct.

Suppose $n\gt3$. Let $G$ be such a graph. There are two cases.

  • There is one node with exactly one edge from it or to it removed.
    Let that node be $u$. By induction hypothesis, there is one Hamiltonian cycle for the induced subgraph of the remaining nodes. Verify that cycle can be modified to pass $u$ as well, hence becoming a Hamiltonian cycle of $G$.
  • Otherwise, for each node, either no edge from it or to it are removed, or at least two edges from it or to it are removed.
    Let $v$ be a node of the former kind and $w$ be a node of the latter kind. Let $G'$ be the induced subgraph of the remaining $n-2$ nodes. Since $2(n-2)\gt n-2$ and there are $2(n-2)$ possible edges between $w$ and a node in $G'$, there must be one edge of $G'$ that is between $w$ and some node of $G'$. By induction hypothesis, $G'$ contains a Hamiltonian cycle. Verify $C$ can be modified to include that edge as well as pass $v$, becoming a Hamiltonian cycle of $G$.

Explanation of Yuval's neat answer

Consider all (directed) Hamiltonian cycles in $K_n$. What is the total number of edges in them, with duplicity counted?

  • Let $f$ be the number of all Hamiltonian cycles. Since each cycle contains $n$ edges, that total number is $nf$.
  • The number of times an edge appearing in those cycles is the same for each edge, thanks to symmetry. Denote it by $p$. Since there are $n(n-1)$ distinct edges, that total number is $n(n-1)p$.

We have, $$ nf = n(n-1)p,\ \ \text{ i.e., }\ \ f= (n-1)p $$

Let us remove edges from $K_n$ so as to obtain the given graph $G$. Since removing an edge affects only Hamiltonian cycles in which that edge appears, removing $n-2$ edges will affect at most $(n-2)p$ Hamiltonian cycles. Since $f=(n-1)p > (n-2)p$, at least one Hamiltonian cycle will not be affected after removing $n-2$ edges. That is, there is at least one Hamiltonian cycle in $G$. $\quad\checkmark$

Stating the explanation in terms of probability and expectation, we shall obtain Yuval's answer.

The only facts about Hamiltonian cycle used in this proof are that it has $n$ edges and that the concept is symmetric to each edge. We have, in fact, proved the following remarkable proposition.

Given $n\ge2$, digraph $G$ of $n$ nodes with $n-2$ edges removed and digraph $D$ of $n$ nodes with $n$ edges, $G$ must contain a subgraph that is isomorphic to $D$.

  • $\begingroup$ This is great! Simple and comprehensive. Thank you! $\endgroup$ – Amal Sailendran Jun 8 at 22:38

If you choose a Hamiltonian cycle at random, the expected number of edges missing is strictly less than $1$.

  • $\begingroup$ $\quad$Neat!$\quad$ $\endgroup$ – John L. Jun 6 at 16:50
  • $\begingroup$ That's a completely different method, using elementary probability theory. $\endgroup$ – Yuval Filmus Jun 6 at 17:49

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