212 research outputs found

    The cycle-complete graph Ramsey number r(C8;K8)

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    The Ramsey number R(r,b)R(r, b) is the least positive integer such that every edge 2-coloring of the complete graph KR(r,b)K_{R(r, b)} with colors red and blue either embeds a red KrK_r or a blue KbK_b. We explore various methods to find lower bounds on R(r,b)R(r,b), 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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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 NPNP-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 Fe(3,3;4)F_e(3,3;4), defined as the smallest order of a K4K_4-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 19leqFe(3,3;4)leq94119 leq F_e(3,3;4) leq 941. We improve the upper bound to Fe(3,3;4)leq786F_e(3,3;4) leq 786 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 R(C4,Km)R(C_4,K_m), which is the smallest nn such that any nn-vertex graph contains a cycle of length four or an independent set of order mm. With the help of combinatorial algorithms, we determine R(C4,K9)=30R(C_4,K_9)=30 and R(C4,K10)=36R(C_4,K_{10})=36 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
    corecore