All Trees are Seven-Cordial

TL;DR

This paper proves that all trees are 7-cordial by extending previous results that confirmed this property for smaller values of k, using an adjustment to Hovey's test.

Contribution

It demonstrates that all trees are 7-cordial, completing the sequence of proofs for k up to 7, building on prior work for smaller k values.

Findings

All trees are 7-cordial.

Extension of Hovey's test confirms the property for k=7.

Completes the proof for all trees being k-cordial for k up to 7.

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 edge-weights as the sum modulo $k$ of the labels on incident vertices to a given edge, which furthermore satisfies the following conditions: (i) Each label appears on at most one more vertex than any other label. (ii) Each edge-weight appears on at most one more edge than any other edge-weight. Mark Hovey (1991) conjectured that all trees are $k$-cordial for any integer $k$. Cahit (1987) had shown earlier that all trees are $2$-cordial and Hovey proved that all trees are $3,4,$ and $5$-cordial. Driscoll, et. al. (2017), used an adjustment to Hovey's test to show that all trees are $6$-cordial. It is shown here that all trees are $7$-cordial by that same adjustment.