Path-cordial abelian groups

TL;DR

This paper explores the extension of Hovey's conjecture on $A$-cordial labelings from trees to all path graphs for finite abelian groups, providing partial characterizations and establishing results for specific group classes.

Contribution

It initiates the study of $A$-cordial labelings for path graphs over finite abelian groups beyond cyclic groups, proposing a conjecture and proving it for certain groups.

Findings

Conjecture characterizes groups where all path graphs are $A$-cordial.

Proved the conjecture for various infinite families of groups.

Confirmed the conjecture for all groups of small order.

Abstract

A labeling of the vertices of a graph by elements of any abelian group $A$ induces a labeling of the edges by summing the labels of their endpoints. Hovey defined the graph $G$ to be $A$-cordial if it has such a labeling where the vertex labels and the edge labels are both evenly-distributed over $A$ in a technical sense. His conjecture that all trees $T$ are $A$-cordial for all cyclic groups $A$ remains wide open, despite significant attention. Curiously, there has been very little study of whether Hovey's conjecture might extend beyond the class of cyclic groups. We initiate this study by analyzing the larger class of finite abelian groups $A$ such that all path graphs are $A$-cordial. We conjecture a complete characterization of such groups, and establish this conjecture for various infinite families of groups as well as for all groups of small order.