Matrices of zeros and ones with fixed row and column sum vectors

AbstractLet m and n be positive integers, and let R=(r1,…,rm) and S=(s1,…,sn) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matrices of 0's and 1's having ri1's in row i andsi 1's in column j. A number of new results are proved. The results can be also be formulated in terms of a set of bipartite graps with bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S

Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

Fuzzy Algebraic Theories and M,N-adhesive categories

This thesis deals with two quite unrelated subjects in Computer Science: one is the relationship between algebraic theories and monads, the other one is the study of adhesivity properties of categories. The first part of the thesis begins by revisiting some basic facts regarding monads. Specifically, we review the correspondence between monads, with rank, on the category of sets and functions, and algebraic theories in which the operations’ arity is bounded by some regular cardinal. Next, we move to the heart of this part of the thesis: the extension of this correspondence to the category Fuz(H) of fuzzy sets. This result is obtained by means of a formal system for fuzzy algebraic reasoning. We define a sequent calculus based on two types of propositions: those that establish the equality of terms, and those that assert the membership degree of such terms. We establish a sound semantics for this calculus, and demonstrate the existence of a notion of free model for any theory in the system. This, in turn, allows us, to prove a completeness result: a formula is derivable from a given theory if and only if it is satisfied by all models of the theory. Moreover, we also prove that, under certain restrictions, it is possible to recover models of a given theory as Eilenberg-Moore algebras for a monad on Fuz(H). Finally, leveraging the work of Milius and Urbat, we provide a HSP-like characterizations of subcategories of algebras that are categories of models of specific types of theories. The second part of the thesis is devoted to the study of adhesivity properties of various categories. Adhesive and quasiadhesive categories, and other generalizations such as M,N-adhesive ones, marked a watershed moment for the algebraic approaches to the rewriting of graph-like structures, since they provide an abstract framework where many general results (on, e.g., parallelism) could be recast and uniformly proved. However, often checking that a model satisfies the adhesivity properties is far from immediate. After having recalled, the basic definitions, we present a new criterion giving a sufficient condition for M,N-adhesivity. It is known that in a quasiadhesive category the join of any two regular subobjects is also a regular subobject. Conversely, if regular monomorphisms are adhesive, the existence of a regular join for every pair of regular subobjects implies quasiadhesivity. Furthermore, (quasi)adhesive categories can be embedded in a Grothendieck topos via a functor that preserves pullbacks and pushouts along (regular) monomorphisms. In this paper, we extend these results to M,N-adhesive categories. To achieve this, we introduce the concept of an N-(pre)adhesive morphism, which enables us to express M,N-adhesivity as a condition on the poset of subobjects. Additionally, N-adhesive morphisms allow us to demonstrate how a M,N-adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and M,N-pushouts. Finally, we exploit the previous results to establish adhesivity properties of several existing categories of graph-like structures, including hypergraphs, various kinds of hierarchical graphs (a formalism that is notoriously difficult to fit in the mould of algebraic approaches to rewriting), and combinations of them

Aspects of graph colouring

The four-colour conjecture of 1852, and the total colouring conjecture of 1965, have sparked off many new concepts and conjectures. In this thesis we investigate many of the outstanding conjectures, establishing various related results, and present many conjectures of our own. We give a brief historical introduction (Chapter 1) and establish some notation, terminology and techniques (Chapter 2). Next, in Chapter 3, we examine the use of latin squares to represent edge and total colourings. In Chapters 4 - 6 we deal with vertex, edge and total colourings respectively. Various ways of measuring different aspects of graphs are presented, in particular, the ‘colouring difference’ between two edge-colourings of a graph (Chapter 5) and the ‘beta parameter’ (defined in Chapter 2 and used in Chapters 3 and 6); this is a measure of how far from a type 1 graph a type 2 graph can be. In Chapter 6 we derive an upper bound for the beta value of any near type 1 graph and give the exact results for all Kn. The number of ways of colouring Kn and Kn,,n are also quantified. Chapter 6 also examines Hilton’s concept of conformability. It is shown that every graph with at least A spines is conformable, and an extension to the concept, which we call G*-conformability, is introduced. We then give new necessary conditions for a cubic graph to be type 1 in relation to G*-conformability. Various methods of manipulating graphs are considered and we present: a method to compatibly triangulate a graph G-e; a method of introducing a fourth colour thus allowing a sequence of Kempe interchanges from any edge 3-colouring of a cubic graph to any other; and a method to re-colour a near type 1 graph within a certain bound on beta. We end this thesis with a brief discussion on possible practical uses for colouring graphs. A list of the main results and conjectures is given at the end of each chapter, but a short list of the principle theorems proven is given below