60 research outputs found
Balanced supersaturation for some degenerate hypergraphs
A classical theorem of Simonovits from the 1980s asserts that every graph
satisfying must contain copies of . Recently, Morris and
Saxton established a balanced version of Simonovits' theorem, showing that such
has copies of , which
are `uniformly distributed' over the edges of . Moreover, they used this
result to obtain a sharp bound on the number of -free graphs via the
container method. In this paper, we generalise Morris-Saxton's results for even
cycles to -graphs. We also prove analogous results for complete
-partite -graphs.Comment: Changed title, abstract and introduction were rewritte
Bounds for the number of meeting edges in graph partitioning
summary:Let be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that admits a bipartition such that each vertex class meets edges of total weight at least , where is the total weight of edges of size and is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph (i.e., multi-hypergraph), we show that there exists a bipartition of such that each vertex class meets edges of total weight at least , where is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with edges, except for and , admits a tripartition such that each vertex class meets at least edges, which establishes a special case of a more general conjecture of Bollobás and Scott
On several partitioning problems of Bollobás and Scott
AbstractJudicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let G be a hypergraph with mi edges of size i for i=1,2. We show that for any integer k⩾1, V(G) admits a partition into k sets each containing at most m1/k+m2/k2+o(m2) edges, establishing a conjecture of Bollobás and Scott. We also prove that V(G) admits a partition into k⩾3 sets, each meeting at least m1/k+m2/(k−1)+o(m2) edges, which, for large graphs, implies a conjecture of Bollobás and Scott (the conjecture is for all graphs). For k=2, we prove that V(G) admits a partition into two sets each meeting at least m1/2+3m2/4+o(m2) edges, which solves a special case of a more general problem of Bollobás and Scott
A hierarchy of randomness for graphs
AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems
On Sharp Thresholds in Random Geometric Graphs
We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability
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