Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$

Clique Decompositions in Random Graphs via Refined Absorption

We prove that if $p\ge n^{-\frac{1}{3}+\beta}$ for some $\beta > 0$, then asymptotically almost surely the binomial random graph $G(n,p)$ has a $K_3$-packing containing all but at most $n + O(1)$ edges. Similarly, we prove that if $d \ge n^{\frac{2}{3}+\beta}$ for some $\beta > 0$ and $d$ is even, then asymptotically almost surely the random $d$-regular graph $G_{n,d}$ has a triangle decomposition provided $3 \mid d \cdot n$. We also show that $G(n,p)$ admits a fractional $K_3$-decomposition for such a value of $p$. We prove analogous versions for a $K_q$-packing of $G(n,p)$ with $p\ge n^{-\frac{1}{q+0.5}+\beta}$ and leave of $(q-2)n+O(1)$ edges, for $K_q$-decompositions of $G_{n,d}$ with $(q-1)~|~d$ and $d\ge n^{1-\frac{1}{q+0.5}+\beta}$ provided $q\mid d\cdot n$, and for fractional $K_q$-decompositions.Comment: 49 page

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $\chi_2$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere $S^3$ with face vector $f=(167,1579,2824,1412)$ and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

A theory of flow network typings and its optimization problems

Many large-scale and safety critical systems can be modeled as flow networks. Traditional approaches for the analysis of flow networks are whole-system approaches in that they require prior knowledge of the entire network before an analysis is undertaken, which can quickly become intractable as the size of network increases. In this thesis we study an alternative approach to the analysis of flow networks, which is modular, incremental and order-oblivious. The formal mechanism for realizing this compositional approach is an appropriately defined theory of network typings. Typings are formalized differently depending on how networks are specified and which of their properties is being verified. We illustrate this approach by considering a particular family of flow networks, called additive flow networks. In additive flow networks, every edge is assigned a constant gain/loss factor which is activated provided a non-zero amount of flow enters that edge. We show that the analysis of additive flow networks, more specifically the max-flow problem, is NP-hard, even when the underlying graph is planar. The theory of network typings gives rise to different forms of graph decomposition problems. We focus on one problem, which we call the graph reassembling problem. Given an abstraction of a flow network as a graph G = (V,E), one possible definition of this problem is specified in two steps: (1) We cut every edge of G into two halves to obtain a collection of |V| one-vertex components, and (2) we splice the two halves of all the edges, one edge at a time, in some order that minimizes the complexity of constructing a typing for G, starting from the typings of its one-vertex components. One optimization is minimizing “maximum” edge-boundary degree of components encountered during the reassembling of G (denoted as α measure). Another is to minimize the “sum” of all edge-boundary degrees encountered during this process (denoted by β measure). Finally, we study different variations of graph reassembling (with respect to minimizing α or β) and their relation with problems such as Linear Arrangement, Routing Tree Embedding, and Tree Layout

Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces