191 research outputs found
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
4−Equitable Tree Labelings
We assign the labels {0,1,2,3} to the vertices of a graph; each edge is assigned the absolute difference of the incident vertices’ labels. For the labeling to be 4−equitable, we require the edge labels and vertex labels to each be distributed as uniformly as possible.
We study 4−equitable labelings of different trees and prove all cater-pillars, symmetric generalized n−stars (or symmetric spiders), and complete n −ary trees for all n ∈ N are 4−equitable
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
Alpha Labeling of Amalgamated Cycles
A graceful labeling of a bipartite graph is an \a-labeling if it has the property that the labels assigned to the vertices of one stable set of the graph are smaller than the labels assigned to the vertices of the other stable set. A concatenation of cycles is a connected graph formed by a collection of cycles, where each cycle shares at most either two vertices or two edges with other cycles in the collection. In this work we investigate the existence of \a-labelings for this kind of graphs, exploring the concepts of vertex amalgamation to produce a family of Eulerian graphs, and edge amalgamation to generate a family of outerplanar graphs. In addition, we determine the number of graphs obtained with copies of the cycle , for both types of amalgamations
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