723 research outputs found
A hypergraph Tur\'an theorem via lagrangians of intersecting families
Let \mc{K}_{3,3}^3 be the 3-graph with 15 vertices and , and 11 edges ,
and . We show
that for large , the unique largest \mc{K}_{3,3}^3-free 3-graph on
vertices is a balanced blow-up of the complete 3-graph on 5 vertices. Our proof
uses the stability method and a result on lagrangians of intersecting families
that has independent interest
Hitting time results for Maker-Breaker games
We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property \mP. We focus on three natural
properties for Maker's graph, namely being -vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the -vertex connectivity game
exactly at the time the random graph process first reaches minimum degree ;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree ; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree . The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
On the inducibility of cycles
In 1975 Pippenger and Golumbic proved that any graph on vertices admits
at most induced -cycles. This bound is larger by a
multiplicative factor of than the simple lower bound obtained by a blow-up
construction. Pippenger and Golumbic conjectured that the latter lower bound is
essentially tight. In the present paper we establish a better upper bound of
. This constitutes the first progress towards proving
the aforementioned conjecture since it was posed
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton
cycle, which is applicable to a wide variety of graphs, including relatively
sparse graphs. In contrast to previous criteria, ours is based on only two
properties: one requiring expansion of ``small'' sets, the other ensuring the
existence of an edge between any two disjoint ``large'' sets. We also discuss
applications in positional games, random graphs and extremal graph theory.Comment: 19 page
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