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All trees are six-cordial
For any integer , a tree is -cordial if there exists a labeling
of the vertices of by , inducing a labeling on the edges with
edge-weights found by summing the labels on vertices incident to a given edge
modulo so that each label appears on at most one more vertex than any other
and each edge-weight appears on at most one more edge than any other.
We prove that all trees are six-cordial by an adjustment of the test proposed
by Hovey (1991) to show all trees are -cordial.Comment: 16 pages, 12 figure
On the number of unlabeled vertices in edge-friendly labelings of graphs
Let be a graph with vertex set and edge set , and be a
0-1 labeling of so that the absolute difference in the number of edges
labeled 1 and 0 is no more than one. Call such a labeling
\emph{edge-friendly}. We say an edge-friendly labeling induces a \emph{partial
vertex labeling} if vertices which are incident to more edges labeled 1 than 0,
are labeled 1, and vertices which are incident to more edges labeled 0 than 1,
are labeled 0. Vertices that are incident to an equal number of edges of both
labels we call \emph{unlabeled}. Call a procedure on a labeled graph a
\emph{label switching algorithm} if it consists of pairwise switches of labels.
Given an edge-friendly labeling of , we show a label switching algorithm
producing an edge-friendly relabeling of such that all the vertices are
labeled. We call such a labeling \textit{opinionated}.Comment: 7 pages, accepted to Discrete Mathematics, special issue dedicated to
Combinatorics 201
Some Investigations in the Theory of Graphs
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