Embedding maximal cliques of sets in maximal cliques of bigger sets

AbstractCharacterizations are obtained of the maximal (k + s)-cliques that contain a given maximal k-clique as a substructure: (1) when s = 1; (2) for arbitrary s when no line of the clique contains exactly one point of the subclique. These characterizations are used to obtain maximal cliques which have fewer lines (for given k) than previously known examples

The Ramsey numbers of squares of paths and cycles

The square G 2 of a graph G is the graph on V (G) with a pair of vertices uv an edge whenever u and v have distance 1 or 2 in G. Given graphs G and H, the Ramsey number R(G, H) is the minimum N such that whenever the edges of the complete graph K N are coloured with red and blue, there exists either a red copy of G or a blue copy of H. We prove that for all sufficiently large n we have (Formula presented). We also show that for every γ > 0 and ∆ there exists β > 0 such that the following holds: If G can be coloured with three colours such that all colour classes have size at most n, the maximum degree of G is at most ∆, and G has bandwidth at most βn, then R(G, G) ≤ (3 + γ)n

GraphMineSuite: Enabling High-Performance and Programmable Graph Mining Algorithms with Set Algebra

We propose GraphMineSuite (GMS): the first benchmarking suite for graph mining that facilitates evaluating and constructing high-performance graph mining algorithms. First, GMS comes with a benchmark specification based on extensive literature review, prescribing representative problems, algorithms, and datasets. Second, GMS offers a carefully designed software platform for seamless testing of different fine-grained elements of graph mining algorithms, such as graph representations or algorithm subroutines. The platform includes parallel implementations of more than 40 considered baselines, and it facilitates developing complex and fast mining algorithms. High modularity is possible by harnessing set algebra operations such as set intersection and difference, which enables breaking complex graph mining algorithms into simple building blocks that can be separately experimented with. GMS is supported with a broad concurrency analysis for portability in performance insights, and a novel performance metric to assess the throughput of graph mining algorithms, enabling more insightful evaluation. As use cases, we harness GMS to rapidly redesign and accelerate state-of-the-art baselines of core graph mining problems: degeneracy reordering (by up to >2x), maximal clique listing (by up to >9x), k-clique listing (by 1.1x), and subgraph isomorphism (by up to 2.5x), also obtaining better theoretical performance bounds