Perfect 1-factorisations of circulants with small degree

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed

On the perfect 1-factorisation problem for circulant graphs of degree 4

A 1-factorisation of a graph G is a partition of the edge set of G into 1 factors (perfect matchings); a perfect 1-factorisation of G is a 1-factorisation of G in which the union of any two of the 1-factors is a Hamilton cycle in G. It is known that for bipartite 4-regular circulant graphs, having order 2 (mod 4) is a necessary (but not sufficient) condition for the existence of a perfect 1-factorisation. The only known non-bipartite 4-regular circulant graphs that admit a perfect 1-factorisation are trivial (on 6 vertices). We prove several construction results for perfect 1-factorisations of a large class of bipartite 4-regular circulant graphs. In addition, we show that no member of an infinite family of non-bipartite 4-regular circulant graphs admits a perfect 1-factorisation. This supports the conjecture that there are no perfect 1-factorisations of any connected non-bipartite 4-regular circulant graphs of order at least 8

Small Partial Latin Squares that Cannot be Embedded in a Cayley Table

We answer a question posed by Dénes and Keedwell that is equivalent to the following. For each order n what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order n? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley tables

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

Topics in Extremal and Probabilistic Combinatorics.

PhD ThesisThis thesis encompasses several problems in extremal and probabilistic combinatorics. Chapter 1. Tuza's famous conjecture on the saturation number states that for r-uniform hypergraphs F the value sat(F; n)=nr1 converges. I answer a question of Pikhurko concerning the asymptotics of the saturation number for families of hypergraphs, proving in particular that sat(F; n)=nr1 need not converge if F is a family of r-uniform hypergraphs. Chapter 2. Cern y's conjecture on the length of the shortest reset word of a synchronizing automaton is arguably the most long-standing open problem in the theory of nite automata. We consider the minimal length of a word that resets some k-tuple. We prove that for general automata if this is nite then it is nk1 . For synchronizing automata we improve the upper bound on the minimal length of a word that resets some triple. Chapter 3. The existence of perfect 1-factorizations has been studied for various families of graphs, with perhaps the most famous open problem in the area being Kotzig's conjecture which states that even-order complete graphs have a perfect 1-factorization. In my work I focus on another well-studied family of graphs: the hypercubes. I answer almost fully the question of how close (in some particular sense) to perfect a 1-factorization of the hypercube can be. Chapter 4. The k-nearest neighbour random geometric graph model puts vertices randomly in a d-dimensional box and joins each vertex to its k nearest neighbours. I nd signi cantly improved upper and lower bounds on the threshold for connectivity for the k-nearest neighbour graph in high dimensions. ii