3,290 research outputs found
On the graceful polynomials of a graph
Every graph can be associated with a family of homogeneous polynomials, one for every degree, having as many variables as the number of vertices.
These polynomials are related to graceful labellings: a graceful polynomial with all even coefficients is a basic tool, in some cases, for proving
that a graph is non-graceful, and for generating a possibly infinite class of non-graceful graphs. Graceful polynomials also seem interesting in their
own right. In this paper we classify graphs whose graceful polynomial has all even coefficients, for small degrees up to 4. We also obtain some
new examples of non-graceful graphs
Vertex Graceful Labeling-Some Path Related Graphs
Treating subjects as vertex graceful graphs, vertex graceful labeling, caterpillar, actinia graphs, Smarandachely vertex m-labeling
Symmetry within Solutions
We define the concept of an internal symmetry. This is a symmety within a
solution of a constraint satisfaction problem. We compare this to solution
symmetry, which is a mapping between different solutions of the same problem.
We argue that we may be able to exploit both types of symmetry when finding
solutions. We illustrate the potential of exploiting internal symmetries on two
benchmark domains: Van der Waerden numbers and graceful graphs. By identifying
internal symmetries we are able to extend the state of the art in both cases.Comment: AAAI 2010, Proceedings of Twenty-Fourth AAAI Conference on Artificial
Intelligenc
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
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