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    Towards Resistance Sparsifiers

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    We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1+ϵ)(1+\epsilon)-resistance sparsifier of size O~(n/ϵ)\tilde O(n/\epsilon), and conjecture this bound holds for all graphs on nn nodes. In comparison, spectral sparsification is a strictly stronger notion and requires Ω(n/ϵ2)\Omega(n/\epsilon^2) edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers

    Percolation on sparse random graphs with given degree sequence

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    We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that in fact these coincide. As a special case, our results apply to power law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.Comment: 20 page

    Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions

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    In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large nn: (i) [1-factorization conjecture] Suppose that nn is even and D2n/41D\geq 2\lceil n/4\rceil -1. Then every DD-regular graph GG on nn vertices has a decomposition into perfect matchings. Equivalently, χ(G)=D\chi'(G)=D. (ii) [Hamilton decomposition conjecture] Suppose that Dn/2D \ge \lfloor n/2 \rfloor . Then every DD-regular graph GG on nn vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that GG is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of `exceptional' vertices and edges of GG, then we can extend these path systems into an approximate decomposition of GG into Hamilton cycles (or perfect matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the third paper. We have now combined this series into a single publication [arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2 figure

    A Geometric Theory for Hypergraph Matching

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    We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex geometry, and 'divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, we introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. We determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, our main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, we apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical we defer it to a subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical Society. 101 pages. v2: minor changes including some additional diagrams and passages of expository tex
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