Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

De novo Synthesis of Linoleic Acid in Multiple Collembola Species

Many ecological interactions in communities take place between consumers and the organisms they feed on. Continuous surplus of specific nutritional compounds in the diet may lead to evolutionary changes in the metabolic capacity of the consumer, leaving the biosynthesis of such compounds prone to genetic decay and render organisms auxotrophic. A nutrient that is essential to many organisms is the unsaturated fatty acid, linoleic acid (LA; 18:2n-6), which is important in the maintenance of cell membrane fluidity and as a precursor for signaling molecules. LA is readily synthesized in bacteria, protozoa and plants, but it was long thought that all animals lack this ability. Although the majority of animals lack the ability for LA biosynthesis, an increasing number of studies have shown that LA is commonly synthesized in arthropods. Here, we investigated a basal hexapod group, Collembola, to shed light on early evolution of LA synthetic ability in arthropods and its relation to dietary composition. We use stable isotope labeling to detect biosynthesis of LA in Collembola fed with C-13-OA oleic acid (OA; 18:1n-9), a precursor of LA. Our data demonstrate that LA biosynthesis is common among Collembola with 10 out of 16 tested species being able to synthesize LA and 4 species lacking this ability. However, we did not find clear evidence for a relationship between LA synthetic ability and the natural diet of species. Thus, the selective pressures underlying LA biosynthesis might be species-specific and further research will shed new light on understanding this evolutionary process