What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Sorption of inorganic nanoparticles in woven cellulose fabrics

Titanium dioxide was deposited from aqueous suspension onto cellulosic surfaces.Titania was sourced from Degussa (P25TM,70:30 anatase:rutile).Dry uptake of particles was shown to be rapid and dominant with one-third of the deposition occurring in less than 30 s and over one-half in the first minute.Isotherms were recorded to compare the rate of titanium deposition on dry and pre-wetted cotton.In the dry case uptake reached a maximum in 30 min whereas in the pre-wetted case the uptake was seen to continue beyond 180 min.A broad trend of higher deposition occurring at lower pH was seen,corresponding to the region where surface charges were opposite and thus attractive.Dry pickup was less significant at high pH.The response to varying ionic strength was complex and was attributed to the combined effect of charge screening,particle aggregation and consequent particle entrapment or occlusion.Titania deposition into the interstices of woven cotton sheets resulted in the formation of inorganic,nanoparticulate skeletons which could be isolated by controlled combustion of the cellulose and thus cotton was suggested to have potential for the templated synthesis of high surface area semiconductor materials.     

Syntheses and reactions of some fluorinated dienes,cyclobutenes and furans

There are many reports¹ of the pyrolysis of fluorinated organic compounds, including the defluorination of cyclic fluorocarbons over iron to give aromatic compounds. Extending this technique we have investigated the flow pyrolysis of some readily accessible unsaturated fluorocarbons, such as I, II, and III, and found these to be synthetically

useful routes to fluorinated dienes, cyclobutenes, and furans. Pyrolyses were carried out using a nitrogen flow over platinum, iron or caesium fluoride heated at 430–700°. The various products can all be rationalized in terms of intermediate allylic radicals, and the solid substrate influences which allylic radicals are formed.We are also investigating the chemistry of those now accessible compounds, such as IV, V, and VI, and some of the preliminary results are described.

For example the fluoride ion induced dimerisation of IV gave two major products VII and VIII via a particular interesting mechanism.

     

The maximum edit distance from hereditary graph properties

For a graph property , the edit distance of a graph G from , denoted , is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying . What is the largest possible edit distance of a graph on n vertices from ? Denote this distance by .A graph property is hereditary if it is closed under removal of vertices. In a previous work, the authors show that for any hereditary property, a random graph essentially achieves the maximal distance from , proving: with high probability. The proof implicitly asserts the existence of such , but it does not supply a general tool for determining its value or the edit distance.In this paper, we determine the values of and for some subfamilies of hereditary properties including sparse hereditary properties, complement invariant properties, (r,s)-colorability and more. We provide methods for analyzing the maximum edit distance from the graph properties of being induced H-free for some graphs H, and use it to show that in some natural cases G(n,1/2) is not the furthest graph. Throughout the paper, the various tools let us deduce the asymptotic maximum edit distance from some well studied hereditary graph properties, such as being Perfect, Chordal, Interval, Permutation, Claw-Free, Cograph and more. We also determine the edit distance of G(n,1/2) from any hereditary property, and investigate the behavior of as a function of p.The proofs combine several tools in Extremal Graph Theory, including strengthened versions of the Szemerédi Regularity Lemma, Ramsey Theory and properties of random graphs.