1,182 research outputs found
On Saturated -Sperner Systems
Given a set , a collection is said to
be -Sperner if it does not contain a chain of length under set
inclusion and it is saturated if it is maximal with respect to this property.
Gerbner et al. conjectured that, if is sufficiently large with respect to
, then the minimum size of a saturated -Sperner system
is . We disprove this conjecture
by showing that there exists such that for every and there exists a saturated -Sperner system
with cardinality at most
.
A collection is said to be an
oversaturated -Sperner system if, for every
, contains more
chains of length than . Gerbner et al. proved that, if
, then the smallest such collection contains between and
elements. We show that if ,
then the lower bound is best possible, up to a polynomial factor.Comment: 17 page
Saturation in the Hypercube and Bootstrap Percolation
Let denote the hypercube of dimension . Given , a spanning
subgraph of is said to be -saturated if it does not
contain as a subgraph but adding any edge of
creates a copy of in . Answering a question of Johnson and Pinto, we
show that for every fixed the minimum number of edges in a
-saturated graph is .
We also study weak saturation, which is a form of bootstrap percolation. A
spanning subgraph of is said to be weakly -saturated if the
edges of can be added to one at a time so that each
added edge creates a new copy of . Answering another question of Johnson
and Pinto, we determine the minimum number of edges in a weakly
-saturated graph for all . More generally, we
determine the minimum number of edges in a subgraph of the -dimensional grid
which is weakly saturated with respect to `axis aligned' copies of a
smaller grid . We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and
Computin
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond
The \emph{choice number} of a graph , denoted , is the minimum
integer such that for any assignment of lists of size to the vertices
of , there is a proper colouring of such that every vertex is mapped to
a colour in its list. For general graphs, the choice number is not bounded
above by a function of the chromatic number.
In this thesis, we prove a conjecture of Ohba which asserts that
whenever . We also prove a
strengthening of Ohba's Conjecture which is best possible for graphs on at most
vertices, and pose several conjectures related to our work.Comment: Master's Thesis, McGill Universit
Effect of Social Media Characteristics on Perceptions of Alcohol-Branded Social Media Content
Background: Alcohol advertising has recently expanded onto social networking sites (SNSs), which allows users to interact with alcohol ads through user engagement features (e.g. Likes or Shares) and user-generated comments. Few studies have evaluated alcohol advertising on SNSs or investigated how ad interactivity may influence ad perceptions. Two inter-related studies were conducted to answer these questions. Methods: In Study 1, alcohol advertising posted on Facebook by Budweiser and Bud Light were evaluated for compliance with a self-regulated alcohol advertising code and for thematic content. User-generated comments written in response to these ads were also evaluated. The results of this study were used to inform Study 2, a 2(within) x 2(between) x 2(between) factorial randomized trial. Participants (n=120) viewed four pre-selected Facebook ads, of which two were compliant and two were non-compliant with a self-regulated alcohol advertising code. Participants also viewed real-world high or low user engagement values and real-world pro- or anti-drinking user-generated comments. Ad appeal, drinking intentions, and individual user engagement were measured after viewing each ad. The results were analyzed using hierarchical linear models. Results: In Study 1, 82% of the ads contained 1 or more violations of a self-regulated alcohol advertising code, and 78% of the ads contained one or more content areas previously associated with code violations. Forty-seven percent of the user-generated comments were positive towards the product or drinking. In Study 2, ads non-compliant with a self-regulated alcohol advertising code scored higher on emotional appeal (p=0.004) while compliant ads scored higher on informational (p\u3c0.001) and source appeal (p=0.034). Pro-drinking user-generated comments significantly increased drinking intentions and individual user engagement. Discussion: Self-regulation has failed to prevent potentially harmful content from appearing in Budweiser and Bud Light advertising posted on Facebook. Non-compliance with existing self-regulated alcohol advertising codes was associated with increased emotional appeal, which may result in the ads being remembered more often and recalled more swiftly. Pro-drinking user-generated comments were associated with increased drinking intentions and increased individual user engagement, both of which are associated with increased alcohol consumption. New regulations may be needed to limit alcohol ad content and the influence of user-generated comments
Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected
plane graph with maximum face size D can be colored using at most D+2 colors in
such a way that no face is incident with two vertices of the same color. The
conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for
D=16 and D=17
Beyond Ohba's Conjecture: A bound on the choice number of -chromatic graphs with vertices
Let denote the choice number of a graph (also called "list
chromatic number" or "choosability" of ). Noel, Reed, and Wu proved the
conjecture of Ohba that when . We
extend this to a general upper bound: . Our result is sharp for
using Ohba's examples, and it improves the best-known
upper bound for .Comment: 14 page
- …