2,556 research outputs found
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
A new property of the Lov\'asz number and duality relations between graph parameters
We show that for any graph , by considering "activation" through the
strong product with another graph , the relation between the independence number and the Lov\'{a}sz number of
can be made arbitrarily tight: Precisely, the inequality
becomes asymptotically an equality for a suitable sequence of ancillary graphs
.
This motivates us to look for other products of graph parameters of and
on the right hand side of the above relation. For instance, a result of
Rosenfeld and Hales states that with the fractional
packing number , and for every there exists that makes the
above an equality; conversely, for every graph there is a that attains
equality.
These findings constitute some sort of duality of graph parameters, mediated
through the independence number, under which and are dual
to each other, and the Lov\'{a}sz number is self-dual. We also show
duality of Schrijver's and Szegedy's variants and
of the Lov\'{a}sz number, and explore analogous notions for the chromatic
number under strong and disjunctive graph products.Comment: 16 pages, submitted to Discrete Applied Mathematics for a special
issue in memory of Levon Khachatrian; v2 has a full proof of the duality
between theta+ and theta- and a new author, some new references, and we
corrected several small errors and typo
Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number
We investigate vector chromatic number, Lovasz theta of the complement, and
quantum chromatic number from the perspective of graph homomorphisms. We prove
an analog of Sabidussi's theorem for each of these parameters, i.e. that for
each of the parameters, the value on the Cartesian product of graphs is equal
to the maximum of the values on the factors. We also prove an analog of
Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value
on the categorical product of graphs is equal to the minimum of its values on
the factors. We conjecture that the analogous results hold for vector and
quantum chromatic number, and we prove that this is the case for some special
classes of graphs.Comment: 18 page
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