All trees are six-cordial

TL;DR

This paper proves that all trees can be labeled in a way that balances vertex labels and edge weights modulo 6, extending the concept of $k$-cordial labelings to all trees.

Contribution

It establishes that every tree is six-cordial, using an adaptation of Hovey's 1991 test for $k$-cordiality.

Findings

All trees are six-cordial.

The proof adapts Hovey's test for $k$-cordiality.

The result extends the class of trees known to be $k$-cordial.

Abstract

For any integer $k>0$, a tree $T$ is $k$-cordial if there exists a labeling of the vertices of $T$ by $\mathbb{Z}_k$, inducing a labeling on the edges with edge-weights found by summing the labels on vertices incident to a given edge modulo $k$ so that each label appears on at most one more vertex than any other and each edge-weight appears on at most one more edge than any other. We prove that all trees are six-cordial by an adjustment of the test proposed by Hovey (1991) to show all trees are $k$-cordial.