On Saturated kk-Sperner Systems

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$. A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.Comment: 17 page

Saturation in the Hypercube and Bootstrap Percolation

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. 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 $H$, the reconfiguration problem for homomorphisms to $H$ (also called $H$-colourings) asks: given two $H$-colourings $f$ of $g$ of a graph $G$, is it possible to transform $f$ into $g$ by a sequence of single-vertex colour changes such that every intermediate mapping is an $H$-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs $H$ (e.g. all $C_4$-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever $H$ is a $K_{2,3}$-free quadrangulation of the $2$-sphere (equivalently, the plane) which is not a $4$-cycle. From this result, we deduce an analogous statement for non-bipartite $K_{2,3}$-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and $4$-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs $G$ and $H$ 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 $H$-colourings is PSPACE-complete whenever $H$ is a reflexive $K_{4}$-free triangulation of the $2$-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 $H$-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 $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ 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 $\ch(G)=\chi(G)$ whenever $|V(G)|\leq 2\chi(G)+1$. We also prove a strengthening of Ohba's Conjecture which is best possible for graphs on at most $3\chi(G)$ 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 kk-chromatic graphs with nn vertices

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend this to a general upper bound: $\text{ch}(G)\le \max\{\chi(G),\lceil({|V(G)|+\chi(G)-1})/{3}\rceil\}$. Our result is sharp for $|V(G)|\le 3\chi(G)$ using Ohba's examples, and it improves the best-known upper bound for $\text{ch}(K_{4,\dots,4})$.Comment: 14 page