4 research outputs found
Cycles of a given length in tournaments
We study the asymptotic behavior of the maximum number of directed cycles of
a given length in a tournament: let be the limit of the ratio of the
maximum number of cycles of length in an -vertex tournament and the
expected number of cycles of length in the random -vertex tournament,
when tends to infinity. It is well-known that and . We
show that if and only if is not divisible by four, which
settles a conjecture of Day. If is divisible by four, we show that
and determine the value exactly for
. We also give a full description of the asymptotic structure of
tournaments with the maximum number of cycles of length when is
not divisible by four or .Comment: One of the programs used to verify the validity of Lemma 16 is
available as an ancillary fil
BOUNDING THE NUMBER OF COMPATIBLE SIMPLICES IN HIGHER DIMENSIONAL TOURNAMENTS
A tournament graph G is a vertex set V of size n, together with a directed edge set E ⊂ V × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, j ∈ V and (i, i) ∉ E for all i ∈ V. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the (n − 1)-dimensional simplex on V is given an orientation. In this dissertation we bound the number of compatible k-simplices, that is we bound the number of k-simplices such that its (k − 1)-faces with the already-specified orientation form an oriented boundary. We prove lower and upper bounds for all k ≥ 3. For k = 3 these bounds agree when the number of vertices n is q or q + 1 where q is a prime power congruent to 3 modulo 4. We also prove some lower bounds for values k \u3e 3 and analyze the asymptotic behavior
A Collection of Problems in Extremal Combinatorics
PhDExtremal combinatorics is concerned with how large or small a combinatorial
structure can be if we insist it satis es certain properties. In this thesis we
investigate four different problems in extremal combinatorics, each with its
own unique
flavour.
We begin by examining a graph saturation problem. We say a graph G
is H-saturated if G contains no copy of H as a subgraph, but the addition
of any new edge to G creates a copy of H. We look at how few edges a Kp-
saturated graph can have when we place certain conditions on its minimum
degree.
We look at a problem in Ramsey Theory. The k-colour Ramsey number
Rk(H) of a graph H is de ned as the least integer n such that every k-
colouring of Kn contains a monochromatic copy of H. For an integer r > 3
let Cr denote the cycle on r vertices. By studying a problem related to
colourings without short odd cycles, we prove new lower bounds for Rk(Cr)
when r is odd.
Bootstrap percolation is a process in graphs that can be used to model
how infection spreads through a community. We say a set of vertices in a
graph percolates if, when this set of vertices start off as infected, the whole
graph ends up infected. We study minimal percolating sets, that is, percolating
sets with no proper percolating subsets. In particular, we investigate
if there is any relation between the smallest and the largest minimal percolating
sets in bounded degree graph sequences.
A tournament is a complete graph where every edge has been given
an orientation. We look at the maximum number of directed k-cycles a
tournament can have and investigate when there exist tournaments with
many more k-cycles than expected in a random tournament.Engineering and Physical Sciences Research Council, grant number EP/K50290X/1