Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201

Genome Assembly Techniques

Since the publication of the human genome in 2001, the price and the time of DNA sequencing have dropped dramatically. The genome of many more species have since been sequenced, and genome sequencing is an ever more important tool for biologists. This trend will likely revolutionize biology and medicine in the near future where the genome sequence of each individual person, instead of a model genome for the human, becomes readily accessible. Nevertheless, genome assembly remains a challenging computational problem, even more so with second generation sequencing technologies which generate a greater amount of data and make the assembly process more complex. Research to quickly, cheaply and accurately assemble the increasing amount of DNA sequenced is of great practical importance. In the first part of this thesis, we present two software developed to improve genome assemblies. First, Jellyfish is a fast k-mer counter, capable of handling large data sets. k-mer frequencies are central to many tasks in genome assembly (e.g. for error correction, finding read overlaps) and other study of the genome (e.g. finding highly repeated sequences such as transposons). Second, Chromosome Builder is a scaffolder and contig placement software. It aims at improving the accuracy of genome assembly. In the second part of this thesis we explore several problems dealing with graphs. The theory of graphs can be used to solve many computational problems. For example, the genome assembly problem can be represented as finding an Eulerian path in a de Bruijn graph. The physical interactions between proteins (PPI network), or between transcription factors and genes (regulatory networks), are naturally expressed as graphs. First, we introduce the concept of "exactly 3-edge-connected" graphs. These graphs have only a remote biological motivation but are interesting in their own right. Second, we study the reconstruction of ancestral network which aims at inferring the state of ancestral species' biological networks based on the networks of current species