All trees are six-cordial - Keith Driscoll; Elliot Krop; Michelle Nguyen (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
For any integer k>0, a tree T is k-cordial if there exists a labeling of the vertices of T by 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: enumerateEach label appears on at most one more vertex than any other label.Each edge-weight appears on at most one more edge than any other edge-weight.enumerateMark 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.