Multipartite hypergraphs achieving equality in Ryser's conjecture

A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We also improve the general lower bound on $f(r)$, showing that $f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the $r=3$ case fails for all $r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an $r$-partite hypergraph there exists a set $S$ of size at most $r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac

Comparison of two sowing systems for CTF using commercially available machinery

ArticleThe crop establishment belongs to crucial technology operations. The quality of sowing is the basis for obtaining efficiency of production. Controlled Traffic Farming (CTF) is a technology which prevents excessive soil compaction and minimizes compacted area to the smallest possible area of perman ent traffic lanes (PTL). There were two sowing systems compared, namely row and band sowing when growing winter barley. Sowing parameters as well as all other field operations were identical for both compared systems. Measurements were conducted at an expe rimental field on non - compacted and traffic lane areas where CTF system was introduced in 2009, with 64% of compacted and 36% of non - compacted soil. Six crop parameters were analysed. Generally, it can be concluded that the band sowing performed better in yield (by 9.3% in non - compacted area; by 3.8 % in traffic lane), ear number (by 5.2% in non - compacted area; by 10.1% in traffic lane) and grain number (by 6.3% in non - compacted area; by 8.1% in traffic lane) as well as crop height (by 6.6% in non - compacted area; and by 2.4% in traffic lane). The only parameter performing worse was TGW with decrease of 6.6% in non - compacted area and decrease 2.8% in traffic lane for band system. Differences in number of grain per ear were negligible

Effect of controlled traffic farming on weed occurrence

ArticleSoil compaction caused by field traffic is one of the most important yield limiting factors. Moreover, published results report that soil over-compaction inhibits the uptake of plant nutrients and decreases their ability to compete with weeds. Controlled Traffic Farming (CTF) is technology which prevents excessive soil compaction and minimizes compacted area to the least possible area of permanent traffic lines. A long-term experiment was established at University farm in Kolinany (Slovakia) in 2010 with 6 m OutTrack CTF system. Random Traffic Farming (RTF) is simulated by 1 annual machinery pass crossing the permanent traffic lines. Aim of presented study was to assess the effect of CTF on weed infection pressure. To achieve this, weed occurrence at different traffic treatments was determined. Emerged weeds per square meter were counted, identified and recorded at 14 monitoring points. Results showed that higher weed infection was found at the area with one machinery pass compared to the non-compacted area. Following weeds were identified: Bromus secalinus L., Stellaria media (L.) VILL., Veronica persica POIR. in LAMK., Poa annua L., Polygonum aviculare L., Convolvulus arvensis L. Occurrence of these weeds could be used as soil compaction indicator. Based on these results it can be concluded, that CTF technology has potential to decrease weed infestation in comparison to RTF system due to ration of non-compacted to compacted area. Moreover, with exact localization of weeds in traffic lines together with exact identification of weed species, it is possible to target the application of herbicides

Determining trafficked areas using soil electrical conductivity – a pilot study

ncrease in machinery size and its random traffic at fields cause soil compaction resulting in damage of soil structure and degradation of soil functions. Nowadays, rapid methods to detect soil compaction at fields are of high interest, especially proximal sensing methods such as electrical conductivity measurements. The aim of this work was to investigate whether electromagnetic induction (EMI) could be used to determine trafficked areas in silty clay soil. Results of randomized block experiment showed a high significant difference (p <0.01) in EMI data measured between compacted and non-compacted areas. EMI readings from compacted areas were, on average, 11% (shallow range) and 9% (deep range) higher than non-compacted areas, respectively. This difference was determined in both shallow and deep measuring ranges, indicating that the difference in soil compaction was detected in both topsoil and subsoil. Furthermore, the data was found to have a significant spatial variability, suggesting that, in order to detect the increase in EMI (which shows the increase in soil compaction), data within close surrounding area should be included in the analyses. Correlation coefficient of EMI and penetration resistance (average moisture content 32.5% and 30.8% for topsoil and subsoil) was found to be 0.66

CD-independent subsets in meet-distributive lattices

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Cz\'edli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.Comment: 14 pages, 4 figure

Scanning Phylogenetic Networks is NP-hard

International audiencePhylogenetic networks are rooted directed acyclic graphs used to depict the evolution of a set of species in the presence of reticulate events. Reconstructing these networks from molecular data is challenging and current algorithms fail to scale up to genome-wide data. In this paper, we introduce a new width measure intended to help design faster parameterized algorithms for this task. We study its relation with other width measures and problems in graph theory and finally prove that deciding it is NP-complete, even for very restricted classes of networks