242 research outputs found
On d-graceful labelings
In this paper we introduce a generalization of the well known concept of a
graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling
of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that
{|f(x)-f(y)| | [x,y]\in E(G)}
={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of
d=e we find the classical notion of a graceful labeling and of an odd graceful
labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite
graph G a d-graceful labeling of G with the property that its maximum value on
one of the two bipartite sets does not reach its minimum value on the other
one. We show that these new concepts allow to obtain certain cyclic graph
decompositions. We investigate the existence of d-graceful \alpha-labelings for
several classes of bipartite graphs, completely solving the problem for paths
and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi
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
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