511 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
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
On the structure of path-like trees
We study the structure of path-like trees. In order to do this, we introduce a set of trees that we call expandable trees. In this paper we also generalize the concept of path-like trees and we call such generalization generalized path-like trees. As in the case of path-like trees, generalized path-like trees, have very nice labeling properties
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