Triangles, Long Paths, and Covered Sets

In chapter 2, we consider a generalization of the well-known Maker-Breaker triangle game for uniform hypergraphs in which Maker tries to build a triangle by choosing one edge in each round and Breaker tries to prevent her from doing so by choosing q edges in each round. The main result is the analysis of a new Breaker strategy using potential functions, introduced by Glazik and Srivastav (2019). Both bounds are of the order Θ(n3/2) so they are asymptotically optimal. The constant for the lower bound is 2-o(1) and for the upper bound it is 3√2. In chapter 3, we describe another Maker-Breaker game, namely the P3-game in which Maker tries to build a path of length 3. First, we show that the methods of chapter 2 are not applicable in this scenario and give an intuition why that might be the case. Then, we give a more simple counting argument to bound the threshold bias. In chapter 4, we consider the longest path problem which is a classic NP-hard problem that arises in many contexts. Our motivation to investigate this problem in a big-data context was the problem of genome-assembly, where a long path in a graph that is constructed of the reads of a genome potentially represents a long contiguous sequence of the genome. We give a semi-streaming algorithm. Our algorithm delivers results competitive to algorithms that do not have a restriction on the amount of memory. In chapter 5, we investigate the b-SetMultiCover problem, a classic combinatorial problem which generalizes the set cover problem. Using an LP-relaxation and analysis with the bounded differences inequality of C. McDiarmid (1989), we show that there is a strong concentration around the expectation

Cycle factors and renewal theory

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on $k$-cycles. Here we settle the problem completely: the threshold for a $k$-cycle factor in $G$ as above is $\kappa_0 \log_2 n$ with $\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82$. Precisely, we prove a 2-point concentration result: if $k \geq \kappa_0 \log_2(2n/e)$ divides $n$ then $G$ contains a $k$-cycle factor w.h.p., whereas if $k<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n$ then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph $G(n,p)$. The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page

Automorphisms generating disjoint Hamilton cycles in star graphs

In the first part of the thesis we define an automorphism φn for each star graph Stn of degree n − 1, which yields permutations of labels for the edges of Stn taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known two-labelled Hamilton cycle H1 2(n) in Stn. Our main result is an improvement from the existing lower bound of bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the improvement is from bn/8c to bn/5c. We extend this result to the cases when n is the power of a prime other than 3 and 7. The second part of the thesis studies ‘symmetric’ collections of edge-disjoint Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under general label-mapping automorphisms. We show that, for all even n, there exists a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus, Stn is not symmetrically Hamilton decomposable if n is not prime. We also give cases of even n, in terms of Carmichael’s reduced totient function λ, for which ‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated from H1 2(n) by a single automorphism, can and cannot attain the optimum bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a power of 2, then Stn has a spanning subgraph with more than half of the edges of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains an open problem as to whether the bϕ(n)/2c can be achieved for symmetric collections, but we are able to show that, for certain odd n, a ϕ(n)/4 bound is achievable and optimal for strongly symmetric collections. The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies in computer science. All our results improve on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs

Phylogeographic Study of Apodemus ilex (Rodentia: Muridae) in Southwest China

BACKGROUND: The Mountains of southwest China have complex river systems and a profoundly complex topography and are among the most important biodiversity hotspots in the world. However, only a few studies have shed light on how the mountains and river valleys promote genetic diversity. Apodemus ilex is a fine model for investigating this subject. METHODOLOGY/PRINCIPAL FINDINGS: To assess the genetic diversity and biogeographic patterns of Apodemus ilex, the complete cytochrome b gene sequences (1,140 bp) were determined from 203 samples of A. draco/ilex that were collected from southwest China. The results obtained suggested that A. ilex and A. draco are sistergroups and diverged from each other approximately 2.25 million years ago. A. ilex could be divided into Eastern and Western phylogroups, each containing two sub-groups and being widespread in different geographical regions of the southern Hengduan Mountains and the western Yunnan - Guizhou Plateau. The population expansions of A. ilex were roughly from 0.089 Mya to 0.023 Mya. CONCLUSIONS: Our result suggested that A. ilex is a valid species rather than synonym of A. draco. As a middle-high elevation inhabitant, the phylogenetic pattern of A. ilex was strongly related to the complex geographical structures in southwest China, particularly the existence of deep river valley systems, such as the Mekong and Salween rivers. Also, it appears that the evolutionary history of A. ilex, such as lineage divergences and population expansions were strongly affected by climate fluctuation in the Late Pleistocene

Facilitating Graph Neural Networks with Random Walk on Simplicial Complexes

Node-level random walk has been widely used to improve Graph Neural Networks. However, there is limited attention to random walk on edge and, more generally, on $k$-simplices. This paper systematically analyzes how random walk on different orders of simplicial complexes (SC) facilitates GNNs in their theoretical expressivity. First, on $0$-simplices or node level, we establish a connection between existing positional encoding (PE) and structure encoding (SE) methods through the bridge of random walk. Second, on $1$-simplices or edge level, we bridge edge-level random walk and Hodge $1$-Laplacians and design corresponding edge PE respectively. In the spatial domain, we directly make use of edge level random walk to construct EdgeRWSE. Based on the spectral analysis of Hodge $1$-Laplcians, we propose Hodge1Lap, a permutation equivariant and expressive edge-level positional encoding. Third, we generalize our theory to random walk on higher-order simplices and propose the general principle to design PE on simplices based on random walk and Hodge Laplacians. Inter-level random walk is also introduced to unify a wide range of simplicial networks. Extensive experiments verify the effectiveness of our random walk-based methods.Comment: Accepted by NeurIPS 202

The compositional and evolutionary logic of metabolism

Metabolism displays striking and robust regularities in the forms of modularity and hierarchy, whose composition may be compactly described. This renders metabolic architecture comprehensible as a system, and suggests the order in which layers of that system emerged. Metabolism also serves as the foundation in other hierarchies, at least up to cellular integration including bioenergetics and molecular replication, and trophic ecology. The recapitulation of patterns first seen in metabolism, in these higher levels, suggests metabolism as a source of causation or constraint on many forms of organization in the biosphere. We identify as modules widely reused subsets of chemicals, reactions, or functions, each with a conserved internal structure. At the small molecule substrate level, module boundaries are generally associated with the most complex reaction mechanisms and the most conserved enzymes. Cofactors form a structurally and functionally distinctive control layer over the small-molecule substrate. Complex cofactors are often used at module boundaries of the substrate level, while simpler ones participate in widely used reactions. Cofactor functions thus act as "keys" that incorporate classes of organic reactions within biochemistry. The same modules that organize the compositional diversity of metabolism are argued to have governed long-term evolution. Early evolution of core metabolism, especially carbon-fixation, appears to have required few innovations among a small number of conserved modules, to produce adaptations to simple biogeochemical changes of environment. We demonstrate these features of metabolism at several levels of hierarchy, beginning with the small-molecule substrate and network architecture, continuing with cofactors and key conserved reactions, and culminating in the aggregation of multiple diverse physical and biochemical processes in cells.Comment: 56 pages, 28 figure