Self-complementary graphs and Ramsey numbers Part I: the decomposition and construction of self-complementary graphs

AbstractA new method of studying self-complementary graphs, called the decomposition method, is proposed in this paper. Let G be a simple graph. The complement of G, denoted by Ḡ, is the graph in which V(Ḡ)=V(G); and for each pair of vertices u,v in Ḡ,uv∈E(Ḡ) if and only if uv∉E(G). G is called a self-complementary graph if G and Ḡ are isomorphic. Let G be a self-complementary graph with the vertex set V(G)={v1,v2,…,v4n}, where dG(v1)⩽dG(v2)⩽⋯⩽dG(v4n). Let H=G[v1,v2,…,v2n],H′=G[v2n+1,v2n+2,…,v4n] and H∗=G−E(H)−E(H′). Then G=H+H′+H∗ is called the decomposition of the self-complementary graph G.In part I of this paper, the fundamental properties of the three subgraphs H,H′ and H∗ of the self-complementary graph G are considered in detail at first. Then the method and steps of constructing self-complementary graphs are given. In part II these results will be used to study certain Ramsey number problems (see (II))

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Generalized Paley graphs and their complete subgraphs of orders three and four

Let $k \geq 2$ be an integer. Let $q$ be a prime power such that $q \equiv 1 \pmod {k}$ if $q$ is even, or, $q \equiv 1 \pmod {2k}$ if $q$ is odd. The generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if ${a-b}$ is a $k$-th power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $G_k(q)$, $\mathcal{K}_4(G_k(q))$, which holds for all $k$. This generalizes the results of Evans, Pulham and Sheehan on the original ($k$=2) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in $G_k(q)$, $\mathcal{K}_3(G_k(q))$. In both cases we give explicit determinations of these formulae for small $k$. We show that zero values of $\mathcal{K}_4(G_k(q))$ (resp. $\mathcal{K}_3(G_k(q))$) yield lower bounds for the multicolor diagonal Ramsey numbers $R_k(4)=R(4,4,\cdots,4)$ (resp. $R_k(3)$). We state explicitly these lower bounds for small $k$ and compare to known bounds. We also examine the relationship between both $\mathcal{K}_4(G_k(q))$ and $\mathcal{K}_3(G_k(q))$, when $q$ is prime, and Fourier coefficients of modular forms

Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of R colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.Comment: 29 pages; minor revision

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for R_k(4) and R_k(5) for some small k, including 415 \u3c = R_3(5), 634 \u3c = R_4(4), 2721 \u3c = R_4(5), 3416 \u3c = R_5(4) and 26082 \u3c = R_5(5). Most of the new lower bounds are consequences of general constructions

Low Energy Precision Test of Supersymmetry

Supersymmetry (SUSY) remains one of the leading candidates for physics beyond the Standard Model, and the search for SUSY will be a central focus of future collider experiments. Complementary information on the viability and character of SUSY can be obtained via the analysis of precision electroweak measurements. In this review, we discuss the prospective implications for SUSY of present and future precision studies at low energy.Comment: 118 pages, review pape

Erdos-Hajnal-type theorems in hypergraphs

The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then it must contain a complete or empty tripartite subgraph with parts of order c(log n)^{1/2 + d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the constant d(H), is best possible. We also prove that, for k > 3, no analogue of the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which do not contain cliques or independent sets of size appreciably larger than one would normally expect.Comment: 15 page

Recent developments in graph Ramsey theory

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress