All trees are six-cordial

Keith Driscoll (Department of Mathematics, Clayton State University, 2000 Clayton State Blvd. Morrow, georgia, 30260 USA)
Elliot Krop (Department of Mathematics, Clayton State University, 2000 Clayton State Blvd. Morrow, georgia, 30260 USA)
Michelle Nguyen (Department of Mathematics, Clayton State University, 2000 Clayton State Blvd. Morrow, georgia, 30260 USA)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 5, No 1 (2017): Electronic Journal of Graph Theory and Applications

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.