5,114 research outputs found
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erd\H{o}s--R\'enyi random graph
asks to estimate the probability that the number of
copies of a graph in exceeds its expectation by a factor .
Chatterjee and Dembo showed that in the sparse regime of as
with for an explicit ,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
is a clique. Here we extend the latter work to any fixed graph and
determine a function such that, for as above and any fixed
, the upper tail probability is , where is the maximum degree of . As it turns out, the
leading order constant in the large deviation rate function, , is
governed by the independence polynomial of , defined as where is the number of independent sets of size in . For
instance, if is a regular graph on vertices, then is the
minimum between and the unique positive solution of
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
Another construction of edge-regular graphs with regular cliques
We exhibit a new construction of edge-regular graphs with regular cliques
that are not strongly regular. The infinite family of graphs resulting from
this construction includes an edge-regular graph with parameters . We
also show that edge-regular graphs with -regular cliques that are not
strongly regular must have at least vertices.Comment: 7 page
The existence of designs via iterative absorption: hypergraph -designs for arbitrary
We solve the existence problem for -designs for arbitrary -uniform
hypergraphs~. This implies that given any -uniform hypergraph~, the
trivially necessary divisibility conditions are sufficient to guarantee a
decomposition of any sufficiently large complete -uniform hypergraph into
edge-disjoint copies of~, which answers a question asked e.g.~by Keevash.
The graph case was proved by Wilson in 1975 and forms one of the
cornerstones of design theory. The case when~ is complete corresponds to the
existence of block designs, a problem going back to the 19th century, which was
recently settled by Keevash. In particular, our argument provides a new proof
of the existence of block designs, based on iterative absorption (which employs
purely probabilistic and combinatorial methods).
Our main result concerns decompositions of hypergraphs whose clique
distribution fulfills certain regularity constraints. Our argument allows us to
employ a `regularity boosting' process which frequently enables us to satisfy
these constraints even if the clique distribution of the original hypergraph
does not satisfy them. This enables us to go significantly beyond the setting
of quasirandom hypergraphs considered by Keevash. In particular, we obtain a
resilience version and a decomposition result for hypergraphs of large minimum
degree.Comment: This version combines the two manuscripts `The existence of designs
via iterative absorption' (arXiv:1611.06827v1) and the subsequent `Hypergraph
F-designs for arbitrary F' (arXiv:1706.01800) into a single paper, which will
appear in the Memoirs of the AM
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