1,478 research outputs found
Sparse halves in dense triangle-free graphs
Erd\H{o}s conjectured that every triangle-free graph on vertices
contains a set of vertices that spans at most
edges. Krivelevich proved the conjecture for graphs with minimum degree at
least . Keevash and Sudakov improved this result to graphs with
average degree at least . We strengthen these results by showing
that the conjecture holds for graphs with minimum degree at least
and for graphs with average degree at least for some absolute . Moreover, we show that the
conjecture is true for graphs which are close to the Petersen graph in edit
distance.Comment: 23 page
Densities of Minor-Closed Graph Families
We define the limiting density of a minor-closed family of simple graphs F to
be the smallest number k such that every n-vertex graph in F has at most
kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting
densities. This set of numbers is countable, well-ordered, and closed; its
order type is at least {\omega}^{\omega}. It is the closure of the set of
densities of density-minimal graphs, graphs for which no minor has a greater
ratio of edges to vertices. By analyzing density-minimal graphs of low
densities, we find all limiting densities up to the first two cluster points of
the set of limiting densities, 1 and 3/2. For multigraphs, the only possible
limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
Characteristics of Small Social Networks
Two dozen networks are analyzed using three parameters that attempt to capture important properties of social networks: leadership L, member bonding B, and diversity of expertise D. The first two of these parameters have antecedents, the third is new. A key part of the analysis is to examine networks at multiple scales by dissecting the entire network into its n subgraphs of a given radius of two edge steps about each of the n nodes. This scale-based analysis reveals constraints on what we have dubbed "cognitive" networks, as contrasted with biological or physical networks. Specifically, "cognitive" networks appear to maximize bonding and diversity over a range of leadership dominance. Asymptotic relations between the bonding and diversity measures are also found when small, nearly complete subgraphs are aggregated to form larger networks. This aggregation probably underlies changes in a regularity among the LBD parameters; this regularity is a U-shaped function of networks size, n, which is minimal for networks around 80 or so nodes
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