Factorization of Graphs

PhD ThesisFor d 1; s 0; a (d; d + s)-graph is a graph whose degrees all lie in the interval fd; d + 1; :::; d + sg. For r 1; a 0; an (r; r+a)-factor of a graph G is a spanning (r; r+a)-subgraph of G. An (r; r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r; r + a)-factors. A graph is (r; r + a)-factorable if it has an (r; r + a)-factorization. For t 1, let (r; s; a; t) be the least integer such that, if d (r; s; a; t), then every (d; d + s)-simple graph G has an (r; r + a)-factorization into x (r; r + a)-factors for at least t di erent values of x. Then we show that, for r 3 odd and a 2 even, (r; s; a; t) = ( r tr+s+1 a + (t 1)r + 1 if t 2, or t = 1 and a < r + s + 1; r if t = 1 and a r + s + 1; Similarily, we have evaluated (r; s; a; t) for all other values of r; s; a and t. We call (r; s; a; t) the simple graph threshold number. A pseudograph is a graph where multiple edges and multiple loops are allowed. A loop counts two towards the degree of the vertex it is on. A multigraph here has no loops. For t 1, let (r; s; a; t) be the least integer such that, if d (r; s; a; t), then every (d; d+s)-pseudograph G has an (r; r+a)-factorization into x (r; r+a)-factors for at least t di erent values of x. We call (r; s; a; t) as the pseudograph threshold number. We have also evaluated (r; s; a; t) for all values of r, s, a and t. Note that for r 3 (r; 0; 1; 1) = 1 meaning that (r; 0; 1; 1) cannot be given a nite value. This study provides various generalisations of Petersen's theorem that \Every 2k-regular graph is 2- factorable".

Free nilpotent and HH-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices

Between primitive and 2-transitive : synchronization and its friends

The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe