9 research outputs found
Guessing Games on Triangle-free Graphs
9 pages, submitted to Electronic Journal of Combinatoric9 pages, submitted to Electronic Journal of CombinatoricThe guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markstr om introduced a method to bound the value of the guessing number from below using the fractional clique number Kf(G). In particular they showed gn(G) >= |V(G)| - Kf(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50
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
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 Games on Undirected Graphs
PhDGuessing games for directed graphs were introduced by Riis for studying multiple unicast
network coding problems. In a guessing game, the players toss generalised die
and can see some of the other outcomes depending on the structure of an underlying
digraph. They later simultaneously guess the outcome of their own die. Their objective
is to find a strategy that maximises the probability that they all guess correctly.
The performance of the optimal strategy for a digraph is measured by the guessing
number.
In general, the existence of an algorithm for computing guessing numbers of a graph
is unknown. In the case of undirected graphs, Christofides and Markstr om defined
a strategy that they conjectured to be optimal. One of the main results of this
thesis is a disproof of this conjecture. In particular, we illustrate an undirected
graph on 10 vertices having guessing number which is strictly larger than the lowerbound
provided by Christofides and Markstr om's method. Moreover, even in case
the undirected graph is triangle-free, we establish counter examples to this conjecture
based on combinatorial objects known as Steiner systems.
The main tool thus far for computing guessing numbers of graphs has been information
theoretic inequalities. Using this method, we are able to derive the guessing
numbers of new families of undirected graphs, which in general cannot be computed
directly using a computer. A new result of the thesis is that Shannon's information
inequalities, which work particularly well for a wide range of graph classes, are not
sufficient for computing the guessing number.
Another contribution of this thesis is a firm answer to the question concerning irreversible
guessing games. In particular, we construct a directed graph G with Shannon
upper-bound that is larger than the same bound obtained when we reverse all edges
of G.
Finally, we initialize a study on noisy guessing game, which is a generalization of
noiseless guessing game defined by Riis.
We pose a few more interesting questions, some of which we can answer and some
which we leave as open problems.
5School of Electronic Engineering and Computer
Science
Observations on graph invariants with the Lovász ϑ-function
This paper delves into three research directions, leveraging the Lovász -function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that , it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász -functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the -function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the -function variant by Schrijver does not possess the property of the Lovász -function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory
Guessing games on triangle-free graphs
The guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markström introduced a method to bound the value of the guessing number from below using the fractional clique cover number kappa_f(G). In particular they showed gn(G) >= |V(G)| - kappa_f(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50