212 research outputs found
The cycle-complete graph Ramsey number r(C8;K8)
The cycle-complete graph Ramsey number
r
(
C
m
,
K
n
)
is the smallest integer
N
such that every graph
G
of order
N
contains a cycle
C
m
on
m
vertices or has independent number
α
(
G
)
≥
n
. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that
r
(
C
m
,
K
n
)
=
(
m
−
1
)
(
n
−
1
)
+
1
for all
m
≥
n
≥
3
(except
r
(
C
3
,
K
3
)
=
6
). In this paper we will present a proof for the conjecture in the case
n
=
m
=
8
.Scopu
The theta-complete graph Ramsey number r(θk, K5); k = 7, 8, 9
Finding the Ramsey number is an important problem of the well-known
family of the combinatorial problems in Ramsey theory. In this work, we investigate
the Ramsey number r(θs, K5) for s = 7, 8, 9 where θs is the set of theta graphs of order
s and K5 is a complete graph of order 5. Our result closed the problem of finding
R(θs, K5) for each s ≥ 6.Scopu
Gallai-Ramsey Number of An 8-Cycle
Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we establish the Gallai-Ramsey number of an 8-cycle for all positive integers
An exploration in Ramsey theory
We present several introductory results in the realm of Ramsey Theory, a subfield of Combinatorics and Graph Theory. The proofs in this thesis revolve around identifying substructure amidst chaos. After showing the existence of Ramsey numbers of two types, we exhibit how these two numbers are related. Shifting our focus to one of the Ramsey number types, we provide an argument that establishes the exact Ramsey number for h(k, 3) for k ≥ 3; this result is the highlight of this thesis. We conclude with facts that begin to establish lower bounds on these types of Ramsey numbers for graphs requiring more substructure
Ramsey Theory
The Ramsey number is the least positive integer such that every edge 2-coloring of the complete graph with colors red and blue either embeds a red or a blue . We explore various methods to find lower bounds on , finding new results on fibrations and semicirculant graphs. Then, generalizing the Ramsey number to graphs other than complete graphs, we flesh out the missing details in the literature on a theorem that completely determines the generalized Ramsey number for cycles
Ramsey numbers for sets of small graphs
AbstractThe Ramsey number r=r(G1-G2-⋯-Gm,H1-H2-⋯-Hn) denotes the smallest r such that every 2-coloring of the edges of the complete graph Kr contains a subgraph Gi with all edges of one color, of a subgraph Hi with all edges of a second color. These Ramsey numbers are determined for all sets of graphs with at most four vertices, and in the diagonal case (m=n,Gi=Hi) for all pairs of graphs, one with at most four and the other with five vertices, so as for all sets of graphs with five vertices
Probing the Electroweak Phase Transition at the LHC
We study the correlation between the value of the triple Higgs coupling and
the nature of the electroweak phase transition. We use an effective potential
approach, including higher order, non-renormalizable terms coming from
integrating out new physics. We show that if only the dimension six operators
are considered, large positive deviations of the triple Higgs coupling from its
Standard Model (SM) value are predicted in the regions of parameter space
consistent with a strong first order electroweak phase transition (SFOEPT). We
also show that at higher orders sizable and negative deviations of the triple
Higgs coupling may be obtained, and the sign of the corrections tends to be
correlated with the order of the phase transition. We also consider a singlet
extension of the SM, which allows us to establish the connection with the
effective field theory (EFT) approach and analyze the limits of its validity.
Furthermore, we study how to probe the triple Higgs coupling from the double
Higgs production at the LHC. We show that selective cuts in the invariant mass
of the two Higgs bosons should be used, to maximize the sensitivity for values
of the triple Higgs coupling significantly different from the Standard Model
one.Comment: 43 pages, 4 figure
Solving Hard Graph Problems with Combinatorial Computing and Optimization
Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are -hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest.
Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number , defined as the smallest order of a -free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were . We improve the upper bound to using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number , which is the smallest such that any -vertex graph contains a cycle of length four or an independent set of order . With the help of combinatorial algorithms, we determine and using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic
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