Metric dimension for random graphs

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph $G(n,r_n)$, $n$ vertices are placed randomly in Euclidean $d$-space and edges are added between any pair of vertices distant at most $r_n$ from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for $G(n,r_n)$ in the thermodynamic limit with $nr_n^d =$ const., and also in the dense limit with $n r_n^d \to \infty$, $r_n \to 0$. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the travelling salesman, spanning tree, matching, bipartite matching and bipartite travelling salesman problems, for a general class of weight functions with at most polynomial growth of order $d-\varepsilon$, under thermodynamic scaling of the distance parameter.Comment: 64 page

A new upper bound for 3-SAT

We show that a randomly chosen 3-CNF formula over n variables with clauses-to-variables ratio at least 4.4898 is, as n grows large, asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was 4.506. The first such bound, independently discovered by many groups of researchers since 1983, was 5.19. Several decreasing values between 5.19 and 4.506 were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.Comment: 20 page

On rigidity, orientability and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G ∈ G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [7]. We show that this appearance of a giant component undergoes a continuous transition for q ≤ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2 + 1-core and 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [7] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

A structural investigation of novel thiophene-functionalized BEDT-TTF donors for application as organic field-effect transistors

Three new unsymmetrical thiophene-functionalized bisIJethylenedithio)tetrathiafulvalene (BEDT-TTF) donors (1–3) have been synthesized, characterised and examined as semiconducting materials for organic field-effect transistor (OFET) devices. The X-ray crystal structures of (1) and (2) reveal both neutral donors pack as dimers with lateral S⋯S contacts. For (1) the molecules are co-facially stacked in a head-to-tail manner with some degree of latitudinal slippage. A device prepared from a crystalline thin film of (1) deposited on unmodified silicon wafer substrate displays a mobility of 5.9 × 10−3 cm2 V−1 s−1 with an on/off ratio of 11. The shorter CH2 linker in (2) results in poorer orbital overlap, likely due to significant longitudinal and latitudinal slippage between molecules in the crystal lattice. As a consequence, no field-effect response was observed for the device fabricated from (2)