13,263 research outputs found
Fully Dynamic Algorithm for Top- Densest Subgraphs
Given a large graph, the densest-subgraph problem asks to find a subgraph
with maximum average degree. When considering the top- version of this
problem, a na\"ive solution is to iteratively find the densest subgraph and
remove it in each iteration. However, such a solution is impractical due to
high processing cost. The problem is further complicated when dealing with
dynamic graphs, since adding or removing an edge requires re-running the
algorithm. In this paper, we study the top- densest-subgraph problem in the
sliding-window model and propose an efficient fully-dynamic algorithm. The
input of our algorithm consists of an edge stream, and the goal is to find the
node-disjoint subgraphs that maximize the sum of their densities. In contrast
to existing state-of-the-art solutions that require iterating over the entire
graph upon any update, our algorithm profits from the observation that updates
only affect a limited region of the graph. Therefore, the top- densest
subgraphs are maintained by only applying local updates. We provide a
theoretical analysis of the proposed algorithm and show empirically that the
algorithm often generates denser subgraphs than state-of-the-art competitors.
Experiments show an improvement in efficiency of up to five orders of magnitude
compared to state-of-the-art solutions.Comment: 10 pages, 8 figures, accepted at CIKM 201
Extremal problems involving forbidden subgraphs
In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs.
In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free -colorable graphs and we find the asymptotic threshold of density of -colorable graphs of large girth when .
In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of -dynamic chromatic number of graphs in terms of other parameters.
In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs.
In particular, we characterize hypergraphs with the maximum number of edges that contain no -regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs
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
Long paths and cycles in random subgraphs of graphs with large minimum degree
For a given finite graph of minimum degree at least , let be a
random subgraph of obtained by taking each edge independently with
probability . We prove that (i) if for a function
that tends to infinity as does, then
asymptotically almost surely contains a cycle (and thus a path) of length at
least , and (ii) if , then
asymptotically almost surely contains a path of length at least . Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking to be the complete graph on vertices.Comment: 26 page
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