All Trees are Seven-Cordial
Keith Driscoll

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 , a tree is -cordial if there exists a labeling of the vertices of by , inducing edge-weights as the sum modulo 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 -cordial for any integer . Cahit (1987) had shown earlier that all trees are -cordial and Hovey proved that all trees are and -cordial. Driscoll, et. al. (2017), used an adjustment to Hovey's test to show that all trees are -cordial. It is shown here that all trees are -cordial by that same adjustment.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · graph theory and CDMA systems · Advanced Graph Theory Research
