Generalized packing designs

Generalized $t$-designs, which form a common generalization of objects such as $t$-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\ {\bf 309} (2009), 4835--4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with $t=2$ and block size $k=3$ or 4.Comment: 38 pages, 2 figures, 5 tables, 2 appendices. Presented at 23rd British Combinatorial Conference, July 201

Existential Closure in Line Graphs

A graph $G$ is $n$-existentially closed if, for all disjoint sets of vertices $A$ and $B$ with $|A\cup B|=n$, there is a vertex $z$ not in $A\cup B$ adjacent to each vertex of $A$ and to no vertex of $B$. In this paper, we investigate $n$-existentially closed line graphs. In particular, we present necessary conditions for the existence of such graphs as well as constructions for finding infinite families of such graphs. We also prove that there are exactly two $2$-existentially closed planar line graphs. We then consider the existential closure of the line graphs of hypergraphs and present constructions for $2$-existentially closed line graphs of hypergraphs.Comment: 13 pages, 2 figure

The Edge-Connectivity of Vertex-Transitive Hypergraphs

A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise this result to hypergraphs and show that every connected linear uniform vertex-transitive hypergraph is maximally edge-connected. We also show that if we relax either the linear or uniform conditions in this generalisation, then we can construct examples of vertex-transitive hypergraphs which are not maximally edge-connected.Comment: 8 page

Distance-Restricted Firefighting on Finite Graphs

In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph $G$ and then $b$ firefighters protect $b$ vertices. On each subsequent turn, the fire spreads to the collective unburnt neighbourhood of all the burning vertices and the firefighters again protect $b$ vertices. Once a vertex has been burnt or protected it remains that way for the rest of the game. We previously introduced the concept of $\textit{distance-restricted firefighting}$ where the firefighters' movement is restricted so they can only move up to some fixed distance $d$ and they may or may not be permitted to move through burning vertices. In this paper we establish the NP-Completeness of the distance-restricted versions of the $\textit{Maximum Vertices Saved}$ problem and present an integer program for solving these problems. In the penultimate section we also discuss some interesting properties of the $\textit{Expected Damage}$ function

The Firebreak Problem

Suppose we have a network that is represented by a graph $G$. Potentially a fire (or other type of contagion) might erupt at some vertex of $G$. We are able to respond to this outbreak by establishing a firebreak at $k$ other vertices of $G$, so that the fire cannot pass through these fortified vertices. The question that now arises is which $k$ vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the $k$ vertices of the firebreak. This is the essence of the {\sc Firebreak} decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant-bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems

Cosmological Perturbations from the Standard Model Higgs

We propose that the Standard Model (SM) Higgs is responsible for generating the cosmological perturbations of the universe by acting as an isocurvature mode during a de Sitter inflationary stage. In view of the recent ATLAS and CMS results for the Higgs mass, this can happen if the Hubble rate during inflation is in the range $(10^{10}- 10^{14})$ GeV (depending on the SM parameters). Implications for the detection of primordial tensor perturbations through the $B$-mode of CMB polarization via the PLANCK satellite are discussed. For example, if the Higgs mass value is confirmed to be $m_h=125.5$ GeV and $m_t, \alpha_s$ are at their central values, our mechanism predicts tensor perturbations too small to be detected in the near future. On the other hand, if tensor perturbations will be detected by PLANCK through the $B$-mode of CMB, then there is a definite relation between the Higgs and top masses, making the mechanism predictive and falsifiable.Comment: 11 pages, 2 figures. Minor corrections and references added to match published versio

Age-Related Changes in MicroRNA Expression and Pharmacogenes in Human Liver

Developmental changes in the liver can significantly impact drug disposition. Due to the emergence of microRNAs (miRNAs) as important regulators of drug disposition gene expression, we studied age-dependent changes in miRNA expression. Expression of 533 miRNAs was measured in 90 human liver tissues (fetal, pediatric [1-17 years], and adult [28-80 years]; n = 30 each). In all, 114 miRNAs were upregulated and 72 were downregulated from fetal to pediatric, and 2 and 3, respectively, from pediatric to adult. Among the developmentally changing miRNAs, 99 miRNA-mRNA interactions were predicted or experimentally validated (e.g., hsa-miR-125b-5p-CYP1A1; hsa-miR-34a-5p-HNF4A). In human liver samples (n = 10 each), analyzed by RNA-sequencing, significant negative correlations were observed between the expression of >1,000 miRNAs and mRNAs of drug disposition and regulatory genes. Our data suggest a mechanism for the marked changes in hepatic gene expression between the fetal and pediatric developmental periods, and support a role for these age-dependent miRNAs in regulating drug disposition