Absolute differences along Hamiltonian paths

TL;DR

This paper proves that for any arithmetic progression, there exists a Hamiltonian path with pairwise distinct absolute differences between consecutive vertices, partially confirming a conjecture by Zhi-Wei Sun.

Contribution

It establishes the existence of such Hamiltonian paths in complete graphs on arithmetic progressions, advancing understanding of difference patterns in graph paths.

Findings

Existence of Hamiltonian paths with distinct absolute differences in arithmetic progressions

Partial proof of Zhi-Wei Sun's conjecture

Extension of difference pattern results to complete graphs

Abstract

Given a set $A$ of real numbers consider the complete graph on the elements of $A$. We prove that if $A$ is an arithmetic progression then for every vertex $a\in A$ there exists an hamiltonian path such that the absolute differences of consecutive vertices are pairwise distinct. This result partially proves a conjecture by Zhi-Wei Sun.