16 research outputs found
Bounded colorings of multipartite graphs and hypergraphs
Let be an edge-coloring of the complete -vertex graph . The
problem of finding properly colored and rainbow Hamilton cycles in was
initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied
since then. Recently it was extended to the hypergraph setting by Dudek, Frieze
and Ruci\'nski. We generalize these results, giving sufficient local (resp.
global) restrictions on the colorings which guarantee a properly colored (resp.
rainbow) copy of a given hypergraph .
We also study multipartite analogues of these questions. We give (up to a
constant factor) optimal sufficient conditions for a coloring of the
complete balanced -partite graph to contain a properly colored or rainbow
copy of a given graph with maximum degree . Our bounds exhibit a
surprising transition in the rate of growth, showing that the problem is
fundamentally different in the regimes and Our
main tool is the framework of Lu and Sz\'ekely for the space of random
bijections, which we extend to product spaces
Unavoidable patterns in locally balanced colourings
Which patterns must a two-colouring of contain if each vertex has at
least red and blue neighbours? In this paper,
we investigate this question and its multicolour variant. For instance, we show
that any such graph contains a -blow-up of an \textit{alternating 4-cycle}
with .Comment: Improved expositio
The Turán Density of Tight Cycles in Three-Uniform Hypergraphs
The Turán density of an -uniform hypergraph , denoted , is the limit of the maximum density of an -vertex -uniform hypergraph not containing a copy of , as . Denote by the -uniform tight cycle on vertices. Mubayi and Rödl gave an “iterated blow-up” construction showing that the Turán density of is at least , and this bound is conjectured to be tight. Their construction also does not contain for larger not divisible by , which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Turán density of for all large not divisible by , showing that indeed . To our knowledge, this is the first example of a Turán density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a -uniform analogue of the statement “a graph is bipartite if and only if it does not contain an odd cycle”
The Tur\'an density of tight cycles in three-uniform hypergraphs
The Tur\'an density of an -uniform hypergraph , denoted
, is the limit of the maximum density of an -vertex
-uniform hypergraph not containing a copy of , as .
Denote by the -uniform tight cycle on
vertices. Mubayi and R\"odl gave an ``iterated blow-up'' construction showing
that the Tur\'an density of is at least , and this bound is conjectured to be tight. Their construction also does
not contain for larger not divisible by , which
suggests that it might be the extremal construction for these hypergraphs as
well. Here, we determine the Tur\'an density of for all
large not divisible by , showing that indeed . To our knowledge, this is the first example of a Tur\'an
density being determined where the extremal construction is an iterated blow-up
construction.
A key component in our proof, which may be of independent interest, is a
-uniform analogue of the statement ``a graph is bipartite if and only if it
does not contain an odd cycle''.Comment: 34 pages, 4 figures (final version accepted to IMRN plus a few
comments in conclusion
Universal lower bound for community structure of sparse graphs
We prove new lower bounds on the modularity of graphs. Specifically, the
modularity of a graph with average degree is
, under some mild assumptions on the degree sequence of
. The lower bound applies, for instance, to graphs
with a power-law degree sequence or a near-regular degree sequence.
It has been suggested that the relatively high modularity of the
Erd\H{o}s-R\'enyi random graph stems from the random fluctuations in
its edge distribution, however our results imply high modularity for any graph
with a degree sequence matching that typically found in .
The proof of the new lower bound relies on certain weight-balanced bisections
with few cross-edges, which build on ideas of Alon [Combinatorics, Probability
and Computing (1997)] and may be of independent interest.Comment: 25 pages, 2 figure