SUMMARY / RELATED TOPICS

Bristle

A bristle is a stiff hair or feather, either on an animal, such as a pig, a plant, or on a tool such as a brush or broom. Synthetic materials such as nylon are used to make bristles in items such as brooms and sweepers. Bristles are used to make brushes for cleaning purposes, as they are abrasive; the bristle brush and the scrub brush are common household cleaning tools used to remove dirt or grease from pots and pans. Bristles are used on brushes other than for cleaning, notably paintbrushes. Bristles are distinguished as unflagged. In cleaning applications, flagged bristles are suited for dry cleaning, unflagged suited for wet cleaning. In painting, flagged bristles yield more application. Bristles are found on pig breeds, instead of fur; because the density is less than with fur, pigs are vulnerable to sunburn. One breed, the Tamworth pig, is endowed with a dense bristle structure such that sunburn damage to skin is minimized. Animals named for their bristles include bristlebirds, the bristle-thighed curlew, the bristle-spined porcupine, the Trinity bristle snail.

Bristles anchor worms to the soil to help them move. Paintbrush Types of Bristle Materials Used for Brushes

Line graph of a hypergraph

The line graph of a hypergraph is the graph whose vertex set is the set of the hyperedges of the hypergraph, with two hyperedges adjacent when they have a nonempty intersection. In other words, the line graph of a hypergraph is the intersection graph of a family of finite sets, it is a generalization of the line graph of a graph. Questions about line graphs of hypergraphs are generalizations of questions about line graphs of graphs. For instance, a hypergraph whose edges all have size k is called k-uniform.. In hypergraph theory, it is natural to require that hypergraphs be k-uniform; every graph is the line graph of some hypergraph, given a fixed edge size k, not every graph is a line graph of some k-uniform hypergraph. A main problem is to characterize those that are, for each k ≥ 3. A hypergraph is linear; every graph is the line graph, not only of some hypergraph, but of some linear hypergraph. Beineke characterized line graphs of graphs by a list of 9 forbidden induced subgraphs. No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any k ≥ 3, Lovász showed there is no such characterization by a finite list if k = 3.

Krausz characterized line graphs of graphs in terms of clique covers. A global characterization of Krausz type for the line graphs of k-uniform hypergraphs for any k ≥ 3 was given by Berge. A global characterization of Krausz type for the line graphs of k-uniform linear hypergraphs for any k ≥ 3 was given by Naik et al.. At the same time, they found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69. Metelsky & Tyshkevich and Jacobson, Kézdy & Lehel improved this bound to 19. At last Skums, Suzdal' & Tyshkevich reduced this bound to 16. Metelsky & Tyshkevich proved that, if k > 3, no such finite list exists for linear k-uniform hypergraphs, no matter what lower bound is placed on the degree. The difficulty in finding a characterization of linear k-uniform hypergraphs is due to the fact that there are infinitely many forbidden induced subgraphs. To give examples, for m > 0, consider a chain of m diamond graphs such that the consecutive diamonds share vertices of degree two.

For k ≥ 3, add pendant edges at every vertex of degree 2 or 4 to get one of the families of minimal forbidden subgraphs of Naik and Shrikhande et al. as shown here. This does not rule out either the existence of a polynomial recognition or the possibility of a forbidden induced subgraph characterization similar to Beineke's of line graphs of graphs. There are some interesting characterizations available for line graphs of linear k-uniform hypergraphs due to various authors under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least k3-2k2+1 in Naik et al. is reduced to 2k2-3k+1 in Jacobson, Kézdy & Lehel and Zverovich to characterize line graphs of k-uniform linear hypergraphs for any k ≥ 3. The complexity of recognizing line graphs of linear k-uniform hypergraphs without any constraint on minimum degree is not known. For k = 3 and minimum degree at least 19, recognition is possible in polynomial time. Skums, Suzdal' & Tyshkevich reduced the minimum degree to 10.

There are conjectures in Naik et al.. Jacoboson et al. Metelsky et Zverovich. Beineke, L. W. "On derived graphs and digraphs", in Sachs, H.. Berge, C. Hypergraphs: Combinatorics of Finite Sets, Amsterdam: North-Holland, MR 1013569. Translated from the French. Bermond, J. C.. "Line graphs of hypergraphs I", Discrete Mathematics, 18: 235–241, doi:10.1016/0012-365X90127-3, MR 0463003. Heydemann, M. C.. "Line graphs of hypergraphs II", Colloq. Math. Soc. J. Bolyai, 18, pp. 567–582, MR 0519291. Krausz, J. "Démonstration nouvelle d'une théorème de Whitney sur Mat. Fiz. Lapok, 50: 75–85, MR 0018403. Lovász, L. "Problem 9", Beitrage zur Graphentheorie und deren Ansendungen, Vortgetragen auf dem international Colloquium in Oberhof, p. 313. Jacobson, M. S.. Metelsky, Yury. CO. Naik, Ranjan N.. Naik, Ranjan N.. Skums, P. V.. "Edge intersection of linear 3-uniform hypergraphs", Discrete Mathematics, 309: 3500–3517, doi:10.1016/j.disc.2007.12.08

Órfãos da Terra

Órfãos da Terra is a Brazilian telenovela produced and broadcast by Rede Globo. It premiered on 2 April 2019, replacing Espelho da Vida, ended on 27 September 2019, replaced by Éramos Seis. Written by Duca Rachid and Thelma Guedes, with the collaboration of Dora Castellar, Aimar Labaki, Carolina Ziskind and Cristina Biscay, it stars Julia Dalavia, Renato Góes, Alice Wegmann, Carmo Dalla Vecchia, Rodrigo Simas, Anajú Dorigon, Emanuelle Araújo and Kaysar Dadour in the main roles. 2019: Rose d'Or Awards Soaps or Telenovelas Official website Órfãos da Terra on IMDb