All trees are 6-cordial
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Authors
Elliot Krop
Keith Driscoll
Michelle Nguyen
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Journal Article, Academic Journal
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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:
\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.