Additive Edge Labelings

TL;DR

This paper investigates conditions under which edge labelings in a graph can be derived from vertex labelings using modular addition, providing a polynomial algorithm to find all solutions and exploring connections to toric ideals.

Contribution

It introduces a polynomial-time algorithm to determine and generate all vertex labelings corresponding to given edge labelings in modular graphs, linking to toric ideals.

Findings

Polynomial algorithm for labelings

Characterization of labelings via toric ideals

Complete solutions for edge-to-vertex labelings

Abstract

Let G=(V,E) be a graph and d a positive integer. We study the following problem: for which labelings f_E: E \to Z_d is there a labeling f_V:V \to Z_d such that f_E(i,j) = f_V(i) + f_V(j) (mod d), for every edge (i,j) in E? We also explore the connections of the equivalent multiplicative version to toric ideals. We derive a polynomial algorithm to answer these questions and to obtain all possible solutions.