19 research outputs found
Perfect codes in 2-valent Cayley digraphs on abelian groups
For a digraph , a subset of is a perfect code if
is a dominating set such that every vertex of is dominated by exactly
one vertex in . In this paper, we classify strongly connected 2-valent
Cayley digraphs on abelian groups admitting a perfect code, and determine
completely all perfect codes of such digraphs
Perfect codes in quintic Cayley graphs on abelian groups
A subset of the vertex set of a graph is called a perfect code
of if every vertex of is at distance no more than one to
exactly one vertex in . In this paper, we classify all connected quintic
Cayley graphs on abelian groups that admit a perfect code, and determine
completely all perfect codes of such graphs
On the independence ratio of distance graphs
A distance graph is an undirected graph on the integers where two integers
are adjacent if their difference is in a prescribed distance set. The
independence ratio of a distance graph is the maximum density of an
independent set in . Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM
J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is
equal to the inverse of the fractional chromatic number, thus relating the
concept to the well studied question of finding the chromatic number of
distance graphs.
We prove that the independence ratio of a distance graph is achieved by a
periodic set, and we present a framework for discharging arguments to
demonstrate upper bounds on the independence ratio. With these tools, we
determine the exact independence ratio for several infinite families of
distance sets of size three, determine asymptotic values for others, and
present several conjectures.Comment: 39 pages, 12 figures, 6 table
Study of the Gromov hyperbolicity constant on graphs
The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it.
In this Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant.
Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators.Programa de Doctorado en IngenierĂa Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva TourĂs Loj