1,503 research outputs found
Odd Harmonious Labeling of Some Graphs
The labeling of discrete structures is a potential area of research due to its
wide range of applications. The present work is focused on one such labeling called odd harmonious labeling
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
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 cordial labeling of hypertrees
Let be a vertex labeling of a hypergraph
. This labeling induces an~edge labeling of defined by
, where the sum is taken modulo . We say that is
-cordial if for all the number of vertices with
label differs by at most from the number of vertices with label and
the analogous condition holds also for labels of edges. If admits a
-cordial labeling then is called -cordial. The existence of
-cordial labelings has been investigated for graphs for decades.
Hovey~(1991) conjectured that every tree is -cordial for every .
Cichacz, G\"orlich and Tuza~(2013) were first to investigate the analogous
problem for hypertrees, that is, connected hypergraphs without cycles. The main
results of their work are that every -uniform hypertree is -cordial for
every and that every hypertree with or odd is -cordial.
Moreover, they conjectured that in fact all hypertrees are -cordial. In this
article, we confirm the conjecture of Cichacz et al. and make a step further by
proving that for every hypertree is -cordial.Comment: 12 page
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