1,246 research outputs found
Variations on Graph Products and Vertex Partitions
In this thesis we investigate two graph products called double vertex graphs and complete double vertex graphs, and two vertex partitions called dominator partitions and rankings. We introduce a new graph product called the complete double vertex graph and study its properties. The complete double vertex graph is a natural extension of the Cartesian product and a generalization of the double vertex graph. We establish many properties of complete double vertex graphs, including results involving the chromatic number of a complete double vertex graph and the characterization of planar complete double vertex graphs. We also investigate the important problem of reconstructing the factors of double vertex graphs and complete double vertex graphs. We reconstruct G from double vertex graphs and complete double vertex graphs for different classes of graphs, including cubic graphs. Next, we look at the properties of dominator partitions of graphs. We characterize minimal dominator partitions of a graph G. This helps us to study the properties of the upper dominator partition number and establish bounds on the upper dominator partition number of different families of graphs, including trees. We also calculate the upper dominator partition number of certain classes of graphs, including paths and cycles, which is surprisingly difficult to calculate. Properties of rankings are studied in this thesis as well. We establish more properties of minimal rankings, including results related to permuting the labels of certain minimal rankings of a graph G. In addition, we investigate rankings of the Cartesian product of two complete graphs, also known as the rook\u27s graph. We establish bounds on the rank number of a rook\u27s graph and calculate its arank number using multiple results we obtain on minimal rankings of a rook\u27s graph
Cartesian product of hypergraphs: properties and algorithms
Cartesian products of graphs have been studied extensively since the 1960s.
They make it possible to decrease the algorithmic complexity of problems by
using the factorization of the product. Hypergraphs were introduced as a
generalization of graphs and the definition of Cartesian products extends
naturally to them. In this paper, we give new properties and algorithms
concerning coloring aspects of Cartesian products of hypergraphs. We also
extend a classical prime factorization algorithm initially designed for graphs
to connected conformal hypergraphs using 2-sections of hypergraphs
A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS
sets and generalized KS sets. We then use projective KS sets to characterize
all graphs for which the chromatic number is strictly larger than the quantum
chromatic number. Here, the quantum chromatic number is defined via a nonlocal
game based on graph coloring. We further show that from any graph with
separation between these two quantities, one can construct a classical channel
for which entanglement assistance increases the one-shot zero-error capacity.
As an example, we exhibit a new family of classical channels with an
exponential increase.Comment: 16 page
Structural Properties of Index Coding Capacity Using Fractional Graph Theory
The capacity region of the index coding problem is characterized through the
notion of confusion graph and its fractional chromatic number. Based on this
multiletter characterization, several structural properties of the capacity
region are established, some of which are already noted by Tahmasbi, Shahrasbi,
and Gohari, but proved here with simple and more direct graph-theoretic
arguments. In particular, the capacity region of a given index coding problem
is shown to be simple functionals of the capacity regions of smaller
subproblems when the interaction between the subproblems is none, one-way, or
complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on
Information Theory (ISIT
Fixing number of co-noraml product of graphs
An automorphism of a graph is a bijective mapping from the vertex set of
to itself which preserves the adjacency and the non-adjacency relations of
the vertices of . A fixing set of a graph is a set of those vertices
of which when assigned distinct labels removes all the automorphisms of
, except the trivial one. The fixing number of a graph , denoted by
, is the smallest cardinality of a fixing set of . The co-normal
product of two graphs and , is a graph having the
vertex set and two distinct vertices are adjacent if is adjacent to
in or is adjacent to in . We define a general
co-normal product of graphs which is a natural generalization of the
co-normal product of two graphs. In this paper, we discuss automorphisms of the
co-normal product of graphs using the automorphisms of its factors and prove
results on the cardinality of the automorphism group of the co-normal product
of graphs. We prove that , for
any two graphs and . We also compute the fixing number of the
co-normal product of some families of graphs.Comment: 13 page
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