The problem concerns proving the existence of a Hamiltonian cycle in the graph ( G_n ), where vertices are ( {1, 2, \ldots, n} ) and edges connect pairs ( (v_1, v_2) ) if ( v_1 + v_2 ) is a perfect square.

Key observations from the user's examples and analysis:

1. Graph Structure: Edges in ( G_n ) are defined by sums that are perfect squares. For each vertex ( m ), its neighbor...

What you are looking at is the “square-sum circle” (or “squareful permutation on a circle”) problem. In your notation, it asks for a Hamiltonian cycle in the graph G_n on {1,…,n} with an edge uv whenever u+v is a perfect square.

What is known

• It is known (and easy to check by computer) that there is no Hamiltonian cycle for n ≤ 31. In particular, n = 31 has no square-sum circle. Your program’s ...

What you have stumbled on is one of the little “folklore’’ problems in Hamiltonian‐graph theory whose answer is, as of today, still unknown. In the literature it is usually called the “square‐sum” or “squareful‐permutation’’ problem:

• Define Gₙ on the vertex‐set {1,2,…,n} by joining i to j precisely when i+j is a perfect square.
• We ask for a Hamiltonian cycle in Gₙ, i.e. a circular ar...

What you have rediscovered is a classical unsolved problem that goes under several different names:

• the Square–sum (or Square-pair) graph
• squareful permutations
• the Square-sum Hamilton cycle problem.

For every n≥1 let

 Gₙ = ( V = {1,…,n}, E = { {a,b} ⊆ V : a+b is a perfect square} ).

The question is:

Does Gₙ contain a Hamiltonian cycle (or at least a Hamiltonian path) for every su...

This is a fascinating problem about finding Hamiltonian cycles in graphs where edges connect vertices whose sum is a perfect square. Let me analyze what we know and discuss the current state of this problem.

Problem Definition

You're looking at graphs where:

• Vertices:
• Edges: if and only if for some integer

The questio...

<think> We are given a graph $G_n$ with vertex set $\{1, 2, \dots, n\}$ and an edge between two vertices if their sum is a perfect square. We wish to show that for sufficiently large $n$, this graph has a Hamiltonian cycle.

The problem is equivalent to arranging the numbers to in a circle such that the sum of every two adjacent numbers is a perfect square.

From the provided code and exam...

This is an excellent question about a well-known and surprisingly difficult problem in recreational mathematics and graph theory. It's sometimes called the "Sum of Squares" circle or attributed to a puzzle popularized by Matt Parker.

Your formalization of the problem is perfect. You are looking for a Hamiltonian cycle in the graph where and edges connect numbers tha...

Understanding the Number Circle Problem

The Problem Setup

This problem involves finding a Hamiltonian cycle in a graph where:

• Vertices are integers from 1 to n:
• Two vertices are connected by an edge if their sum is a perfect square:

The brute force results suggest that...

The graph you defined

  Gₙ = (V,E) ,  V = { 1,…,n },    E = { {u,v} ⊂ V : u+v is a perfect square }

is usually called the square-sum graph or the square graph on the first n positive integers.
Putting the vertices on a circle in such an order that every two neighbours add up to a square is exactly the same as asking whether Gₙ possesses a Hamiltonian cycle (sometimes one only asks for a Hamilt...