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))

Advice Complexity of the Online Induced Subgraph Problem

Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given property of the input graph. Examples of such problems include maximum independent set, maximum planar graph, and many others. We consider these problems, where the vertices are presented online. With each vertex, the online algorithm must decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating the generalized problem: for a hereditary property \pty, find some maximal induced subgraph having \pty. We study this problem from the point of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we give a tight trade-off relationship stating that for inputs of length n roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of \pty, for any c (possibly a function of n). Surprisingly, a similar result cannot be obtained for the symmetric problem: for a given cohereditary property \pty, find a minimum subgraph having \pty. We show that the advice complexity of this problem varies significantly with the choice of \pty. We also consider preemptive online model, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set again. We show that, for the maximum induced subgraph problem, preemption cannot help much, giving a lower bound of $\Omega(n/(c^2\log c))$ bits of advice needed to obtain competitive ratio $c$, where $c$ is any increasing function bounded by \sqrt{n/log n}. We also give a linear lower bound for c close to 1

Hipergráfok = Hypergraphs

A projekt célkitűzéseit sikerült megvalósítani. A négy év során több mint száz kiváló eredmény született, amiből eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyítottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérő, hogy a résztvevők egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele díjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize

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