Some results on (a:b)-choosability

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound for the $k^{th}$ choice number of a graph is given. We also prove that a directed graph with maximum outdegree $d$ and no odd directed cycle is $(k(d+1):k)$-choosable for every $k \geq 1$. Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability

Creating a Norm for the Vernacular

The article deals with the birth of a linguistic norm in Iceland and Italy. The dis­cussion focuses on four works, which lay the foundations for the discussion of grammar and poetics in their respective vernaculars, namely Dante Alighieri’s De vulgari eloquentia and Convivio for Italian, and the First Grammatical Trea­tise and Snorri Sturluson’s Edda for Icelandic. A parallel between these four works is established, and the view that Latin has been little used in Iceland dur­ing the Middle Ages is challenged, also in accordance with both earlier scholar­ship (Lehmann 1937 and Walter 1976) and recent discoveries (Gottskálk Jensson 2002, 2004, 2009 and Marner 2016). It is argued that Latin is bound to have been used as a language of scholarship in Iceland as it was in Western Europe, although manuscript transmission seldom provides direct evidence in this respect. More­over, a view that takes into account the different weight that Latin as such had in the two different speech communities, Italian and Icelandic, is advocated. This approach rests upon the fact that, whereas in Italy there was an unbroken literary tradition in Latin from Roman times to the Middle Ages, in Ice­land Icelandic was the only language to be used until the Conversion, i.e. until the Latin alphabet was introduced. Thus, it is not surprising that the Icelandic ver­nac­u­lar was held in relatively higher esteem in Iceland, therefore leading to a rel­a­tively earlier and richer literary tradition in that language, whereas in Italy the ver­nac­u­lar had to be first raised in linguistic status in order to be used as literary lan­guage.Peer Reviewe

Graphs Where Every Maximal Path Is Maximum

AbstractWe give a complete characterization of all graphs in which every simple path is contained in a maximum path. Among Hamiltonian graphs, such graphs were previously characterized by C. Thomassen

Flows and bisections in cubic graphs

A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$ vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph $G$ with a circular nowhere-zero $r$-flow has a $\lfloor r \rfloor$-weak bisection. In this paper we study problems related to the existence of $k$-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio