4,742 research outputs found
On the hat guessing number of a planar graph class
The hat guessing number is a graph invariant based on a hat guessing game
introduced by Winkler. Using a new vertex decomposition argument involving an
edge density theorem of Erd\H{o}s for hypergraphs, we show that the hat
guessing number of all outerplanar graphs is less than . We also
define the class of layered planar graphs, which contains outerplanar graphs,
and we show that every layered planar graph has bounded hat guessing number.Comment: 13 pages, 2 figures + appendi
Bears with Hats and Independence Polynomials
Consider the following hat guessing game. A bear sits on each vertex of a
graph , and a demon puts on each bear a hat colored by one of colors.
Each bear sees only the hat colors of his neighbors. Based on this information
only, each bear has to guess colors and he guesses correctly if his hat
color is included in his guesses. The bears win if at least one bear guesses
correctly for any hat arrangement.
We introduce a new parameter - fractional hat chromatic number ,
arising from the hat guessing game. The parameter is related to the
hat chromatic number which has been studied before. We present a surprising
connection between the hat guessing game and the independence polynomial of
graphs. This connection allows us to compute the fractional hat chromatic
number of chordal graphs in polynomial time, to bound fractional hat chromatic
number by a function of maximum degree of , and to compute the exact value
of of cliques, paths, and cycles
Extremal/Saturation Numbers for Guessing Numbers of Undirected Graphs
Hat guessing games—logic puzzles where a group of players must try to guess the color of their own hat—have been a fun party game for decades but have become of academic interest to mathematicians and computer scientists in the past 20 years. In 2006, Søren Riis, a computer scientist, introduced a new variant of the hat guessing game as well as an associated graph invariant, the guessing number, that has applications to network coding and circuit complexity. In this thesis, to better understand the nature of the guessing number of undirected graphs we apply the concept of saturation to guessing numbers and investigate the extremal and saturation numbers of guessing numbers. We define and determine the extremal number in terms of edges for the guessing number by using the previously established bound of the guessing number by the chromatic number of the complement. We also use the concept of graph entropy, also developed by Søren Riis, to find a constant bound on the saturation number of the guessing number
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
On 1-factorizations of Bipartite Kneser Graphs
It is a challenging open problem to construct an explicit 1-factorization of
the bipartite Kneser graph , which contains as vertices all -element
and -element subsets of and an edge between any
two vertices when one is a subset of the other. In this paper, we propose a new
framework for designing such 1-factorizations, by which we solve a nontrivial
case where and is an odd prime power. We also revisit two classic
constructions for the case --- the \emph{lexical factorization} and
\emph{modular factorization}. We provide their simplified definitions and study
their inner structures. As a result, an optimal algorithm is designed for
computing the lexical factorizations. (An analogous algorithm for the modular
factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a
odd prime powe
Hypercube orientations with only two in-degrees
We consider the problem of orienting the edges of the -dimensional
hypercube so only two different in-degrees and occur. We show that this
can be done, for two specified in-degrees, if and only if an obvious necessary
condition holds. Namely, there exist non-negative integers and so that
and . This is connected to a question arising from
constructing a strategy for a "hat puzzle."Comment: 9 pages, 4 figure
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