TITLE: Connected multigraph with degree sequence $1, 4, 9, 16, …$
QUESTION [9 upvotes]: An archipelago consists of $n$ islands. Some of the islands are pairwise connected by bridges (maybe zero or more than one). It is known that from any island one can arrive to any other island, and that the first island has $1$ bridge, the second $4$, ..., the $n$-th has $n^2$ bridges.
Find all $n$ for which such an archipelago can exist.

(Belarussian math contest 1992)

REPLY [3 votes]: I found another proof for sufficiently with induction. Since the configurations for a base cases $n=7$ and $n=8$ are easy to find I'll skip to induction step $n \to n+4$:
Say we have $n+4$ islands. Then connect the island
$A_{n+3}$ with $(n+3)^2$ bridges with $A_{n+4}$ and this one with
another $2n+3$ bridges to $A_{n+2}$ and this one with another
$(n+1)^2$ bridges to $A_{n+1}$. Now we have to make $4$ more bridges
from $A_{n+4}$.

By induction hypothesis we can arrange bridges between island
$A_1,A_2,...A_n$. It is not difficult to see that $A_n$ is
connected with some of the island $A_1,A_2,...A_{n-1}$ with at
least $3$ bridges (suppose it isn't true, then $n^2\leq 1+2(n-2)
\Longrightarrow (n-1)^2\leq -2)$. Call this island $B$. Then we
can destroy $2$ bridges between $A_n$ and $B$. Then connect
$A_{n+4}$ with $A_n$ with $2$ new bridge and the same with $B$.