New Computational Result on Harmonious Trees

TL;DR

This paper uses a computational hybrid algorithm to verify that all trees with up to 31 nodes are harmonious, providing new evidence towards Graham and Sloane's conjecture from 1980.

Contribution

It introduces a computational approach and hybrid algorithm to verify the harmonious labelling conjecture for trees up to 31 nodes, extending previous results.

Findings

All trees with at most 31 nodes are harmonious.

Extended the verified size limit for the conjecture.

Demonstrated effectiveness of hybrid algorithms in graph labelling problems.

Abstract

Graham and Sloane proposed in 1980 a conjecture stating that every tree has a harmonious labelling, a graph labelling closely related to additive base. Very limited results on this conjecture are known. In this paper, we proposed a computational approach to this conjecture by checking trees with limited size. With a hybrid algorithm, we are able to show that every tree with at most 31 nodes is harmonious, extending the best previous result in this direction.