Structural properties of graphs and the Universal Difference Property
Combinatorics
2026-01-21 v1
Abstract
We study the Universal Difference Property (UDP) introduced by Alt{\i}nok, Anders, Arreola, Asencio, Ireland, Sar{\i}o\u{g}lan, and Smith, focusing on the relationship between the structural properties of a graph and UDP. We present condtions for when UDP must hold on unicyclic graphs. We then prove that if UDP does not hold on an edge-labeled graph, then it cannot hold on any subdivision of that graph. Additionally, we show that if an edge-labeled graph satisfies the pairwise edge-disjoint path property, then the graph satisfies UDP. Lastly, we explore the relationship between UDP and subgraphs and prove that trees and cycles are the only two families of connected graphs for which UDP must hold for any edge-labeling over any ring.
Cite
@article{arxiv.2601.14126,
title = {Structural properties of graphs and the Universal Difference Property},
author = {Katie Anders and Able Martinez and Patrick McHugh and Jenna Rogers and Remi Salinas Schmeis},
journal= {arXiv preprint arXiv:2601.14126},
year = {2026}
}