All trees are six-cordial - Keith Driscoll; Elliot Krop; Michelle Nguyen (Electronic Journal of Graph Theory and Applications (EJGTA), 2017)
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: \begin{enumerate}\item Each label appears on at most one more vertex than any other label.\item Each edge-weight appears on at most one more edge than any other edge-weight.\end{enumerate}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. We show that all trees are six-cordial by an adjustment of the test proposed by Hovey to show all trees are $k$-cordial.