628 research outputs found
An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles
In 1991, Gnanajothi [4] proved that the path graph P_n with n vertex and n-1
edge is odd graceful, and the cycle graph C_m with m vertex and m edges is odd
graceful if and only if m even, she proved the cycle graph is not graceful if m
odd. In this paper, firstly, we studied the graph C_m P_m when m = 4,
6,8,10 and then we proved that the graph C_ P_n is odd graceful if m is
even. Finally, we described an algorithm to label the vertices and the edges of
the vertex set V(C_m P_n) and the edge set E(C_m P_n).Comment: 9 Pages, JGraph-Hoc Journa
Vertex Graceful Labeling-Some Path Related Graphs
Treating subjects as vertex graceful graphs, vertex graceful labeling, caterpillar, actinia graphs, Smarandachely vertex m-labeling
On the Graceful Game
A graceful labeling of a graph with edges consists of labeling the
vertices of with distinct integers from to such that, when each
edge is assigned as induced label the absolute difference of the labels of its
endpoints, all induced edge labels are distinct. Rosa established two well
known conjectures: all trees are graceful (1966) and all triangular cacti are
graceful (1988). In order to contribute to both conjectures we study graceful
labelings in the context of graph games. The Graceful game was introduced by
Tuza in 2017 as a two-players game on a connected graph in which the players
Alice and Bob take turns labeling the vertices with distinct integers from 0 to
. Alice's goal is to gracefully label the graph as Bob's goal is to prevent
it from happening. In this work, we study winning strategies for Alice and Bob
in complete graphs, paths, cycles, complete bipartite graphs, caterpillars,
prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths
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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Some Topics of Special Interest in Graph Theory
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