5,684 research outputs found
Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems
Friedland's Lower Matching Conjecture asserts that if is a --regular
bipartite graph on vertices, and denotes the number of
matchings of size , then where . When
, this conjecture reduces to a theorem of Schrijver which says that a
--regular bipartite graph on vertices has at least
perfect matchings. L. Gurvits
proved an asymptotic version of the Lower Matching Conjecture, namely he proved
that
In this paper, we prove the Lower Matching Conjecture. In fact, we will prove
a slightly stronger statement which gives an extra factor
compared to the conjecture if is separated away from and , and is
tight up to a constant factor if is separated away from . We will also
give a new proof of Gurvits's and Schrijver's theorems, and we extend these
theorems to --biregular bipartite graphs
Ramanujan Graphs in Polynomial Time
The recent work by Marcus, Spielman and Srivastava proves the existence of
bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that
paper did not provide a polynomial time algorithm to actually compute such
graphs. Here, we provide a polynomial time algorithm to compute certain
expected characteristic polynomials related to this construction. This leads to
a deterministic polynomial time algorithm to compute bipartite Ramanujan
(multi)graphs of all degrees and all sizes
Random graphs containing arbitrary distributions of subgraphs
Traditional random graph models of networks generate networks that are
locally tree-like, meaning that all local neighborhoods take the form of trees.
In this respect such models are highly unrealistic, most real networks having
strongly non-tree-like neighborhoods that contain short loops, cliques, or
other biconnected subgraphs. In this paper we propose and analyze a new class
of random graph models that incorporates general subgraphs, allowing for
non-tree-like neighborhoods while still remaining solvable for many fundamental
network properties. Among other things we give solutions for the size of the
giant component, the position of the phase transition at which the giant
component appears, and percolation properties for both site and bond
percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
Matchings on infinite graphs
Elek and Lippner (2010) showed that the convergence of a sequence of
bounded-degree graphs implies the existence of a limit for the proportion of
vertices covered by a maximum matching. We provide a characterization of the
limiting parameter via a local recursion defined directly on the limit of the
graph sequence. Interestingly, the recursion may admit multiple solutions,
implying non-trivial long-range dependencies between the covered vertices. We
overcome this lack of correlation decay by introducing a perturbative parameter
(temperature), which we let progressively go to zero. This allows us to
uniquely identify the correct solution. In the important case where the graph
limit is a unimodular Galton-Watson tree, the recursion simplifies into a
distributional equation that can be solved explicitly, leading to a new
asymptotic formula that considerably extends the well-known one by Karp and
Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
- âŠ