Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V_1,...,V_t such that for all i the subtournament T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k,t) = O(k^7 t^4) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists an integer h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t) = O(t^5) suffices.Comment: final version, to appear in Combinatoric

Building virtual bridges: how rural micro-enterprises develop social capital in online and face-to-face settings

In rural UK, businesses are often isolated and have much to gain from healthy networks, yet studies show that many rural business owners fail to network effectively. Information communications technologies offer new ways to network that might benefit rural businesses by expanding their reach. This study looked at online and face-to-face networking behaviour among rural micro-enterprises in Scotland in relation to the development of bonding and bridging social capital. Given the challenges of remoteness faced by many rural businesses, online networking is particularly useful in developing bridging capital, but is an unsuitable context for building the trust needed to gain tangible benefits. The article therefore highlights the importance of face-to-face interactions in developing trust and bonding social capital. Rural business owners face distinctive challenges with respect to online communications, which are explored in this article

Hybridization and genetic distances suggest one large monobaramin in the gourd family (Cucurbitales: Cucurbitaceae)

The gourds form a primarily tropical and subtropical family (Cucurbitales: Cucurbitaceae) of herbaceous climbers and woody lianas composed of approximately 960 species in 95 genera and 15 tribes. Many of these are globally important food crops and ornamentals including cucumbers, gourds, squashes, melons, pumpkins, and luffas. Utilizing published literature, there is evidence of interspecific hybridization (natural, experimental cross-pollination, and embryo/ovule culture) in five of the 15 tribes. These consist of successful crosses between 131 unique species pairs, including two intergeneric and one intertribal cross, forming eight monobaramins. In addition, species that do not hybridize directly, but hybridize with the same third species are added to the various monobaramins. This results in an additional 227 species pairs, including two intergeneric and two intertribal. A hybridogram, constructed from these direct and indirect hybridization results, reveals the following monobaramins: 1) 23 species from genus Cucumis; 2) 18 species from genus Cucurbita; 3) nine species from genus Luffa; 4) four species from genera Trichosanthes and Momordica (includes intertribal crosses); 5) three species from genus Citrullus; 6) two species from genus Lagenaria; 7) two species from genus Bryonia; and 8) two species from genera Coccinia and Diplocyclos. Next, using genetic distances (ITS regions of nuclear ribosomal RNA genes) from 26 cucurbit species, several monobaramins are enlarged and some are even connected by adding an additional 113 species pairs, including 82 intergeneric and 58 intertribal. A revised hybridogram, utilizing all of the above methods, reveals the following: 1) the original [Trichosanthes + Momordica] monobaramin connects to the original Luffa and Bryonia monobaramins as well as a new monobaramin, composed of the genera [Echinopepon + Cyclanthera + Sicyos + Ecballium], to form one large monobaramin consisting of 20 species from eight genera and three tribes (Sicyoeae, Bryonieae, and Momordiceae); 2) the original Citrullus monobaramin connects to a member of the genus Lagenaria and the resulting monobaramin is slightly larger with six species; 3) the original Cucumis monobaramin has additional internal continuity and may connect to the contribal [Coccinia + Diplocyclos] and [Citrullus + Lagenaria] monobaramins to form one large monobaramin consisting of 31 species from five genera in Tribe Benincaseae; 4) the original Cucurbita monobaramin connects to one additional species and has 19 species overall, all from genus Cucurbita and Tribe Cucurbiteae. Finally, the [Sicyoeae + Bryonieae + Momordiceae] monobaramin connects directly to both the Cucurbiteae and Benincaseae monobaramins. Connections between these last two tribes are also suggested by grafting and somatic hybridization experiments which are generally most successful between closely related species. In conclusion, these data suggest that members of all five tribes (70 species and 14 genera) may form one large monobaramin in the Family Cucurbitaceae. Additional research is required (hybridization, molecular, morphological) to determine the baraminological status of the family as a whole

BRAF Mutation in Colorectal Cancer

The BRAF mutant colorectal cancer subgroup is a small population with unique clinicopathological and molecular features. This subgroup has been associated with particularly poor prognosis and advanced disease. The poor response of these patients to available treatments has driven much of the effort in trialling combination targeted treatments involving BRAF and MEK inhibitors. Most recently, an observed survival benefit with intensive triplet chemotherapy agents would encourage its use as first-line treatment in suitable candidates given that few of these patients proceed to second- or third-line treatments

New old stones at Antiochia in Rough Cilicia: A novel city name and a proposed visit by Hadrian and Sabina

This article considers the evidence of newly discovered inscriptions from Antiochia ad Cragum in western Rough Cilicia and proposes two distinct observations: one, the city had an additional civic name different from that which is most commonly known; and two, the emperor Hadrian and Sabina may have visited the city and region during their journey from Egypt to Athens in 131 CE. The excavations at the Roman-era city of Antiochia ad Cragum on the Turkish south coast have been ongoing since 2005, and since the beginning inscriptions have been discovered that shed helpful light on the history of the city. However, 2018 was an exceptional year for epigraphical discoveries in which several inscribed stones were found that challenge long-held notions about the city and add significantly to the historical record. Two of the inscriptions found that year offer the first instances on stone of a novel toponymic epithet for the city rather than the commonly held toponymic ad Cragum. The editiones principes of both, dealing solely with physical autopsy and textual reconstruction, have appeared separately.1 The current paper addresses the historical context of these inscriptions – civic nomenclature and possible imperial patronage in particular. The first section considers the formation of the city under Antiochus IV of Commagene and how these new inscriptions and known civic coinage reveal a different name than that by which the city is commonly known. In the second section, epigraphical evidence from one of the inscriptions is used to build a circumstantial case for a proposed visit to the city by Hadrian and Sabina in 131 CE as part of their westward journey from Egypt to Athens

Extremal problems on graphs, directed graphs and hypergraphs

This thesis is concerned with extremal problems on graphs and similar structures. We first study degree conditions in uniform hypergraphs that force matchings of various sizes. Our main result in this area improves bounds of Markstrom and Rucinski on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We then study connectivity conditions in tournaments that ensure the existence of partitions of the vertex set that satisfy various properties. In 1982 Thomassen asked whether every sufficiently strongly connected tournament T admits a partition of its vertex set into t vertex classes such that the subtournament induced on T by each class is strongly k-connected. Our main result in this area implies an affirmative answer to this question. Finally we investigate the typical structure of graphs and directed graphs with some forbidden subgraphs. We answer a question of Cherlin by finding the typical structure of triangle-free oriented graphs. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. We also determine, for all k>5, the typical structure of graphs that do not contain an induced 2k-cycle. This verifies a conjecture of Balogh and Butterfield