41,958 research outputs found
Guessing Numbers of Odd Cycles
For a given number of colours, , the guessing number of a graph is the
base logarithm of the size of the largest family of colourings of the
vertex set of the graph such that the colour of each vertex can be determined
from the colours of the vertices in its neighbourhood. An upper bound for the
guessing number of the -vertex cycle graph is . It is known that
the guessing number equals whenever is even or is a perfect
square \cite{Christofides2011guessing}. We show that, for any given integer
, if is the largest factor of less than or equal to
, for sufficiently large odd , the guessing number of with
colours is . This answers a question posed by
Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also
present an explicit protocol which achieves this bound for every . Linking
this to index coding with side information, we deduce that the information
defect of with colours is for sufficiently
large odd . Our results are a generalisation of the case which was
proven in \cite{bar2011index}.Comment: 16 page
Graph Theory versus Minimum Rank for Index Coding
We obtain novel index coding schemes and show that they provably outperform
all previously known graph theoretic bounds proposed so far. Further, we
establish a rather strong negative result: all known graph theoretic bounds are
within a logarithmic factor from the chromatic number. This is in striking
contrast to minrank since prior work has shown that it can outperform the
chromatic number by a polynomial factor in some cases. The conclusion is that
all known graph theoretic bounds are not much stronger than the chromatic
number.Comment: 8 pages, 2 figures. Submitted to ISIT 201
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