56 research outputs found
Compactness and finite forcibility of graphons
Graphons are analytic objects associated with convergent sequences of graphs.
Problems from extremal combinatorics and theoretical computer science led to a
study of graphons determined by finitely many subgraph densities, which are
referred to as finitely forcible. Following the intuition that such graphons
should have finitary structure, Lovasz and Szegedy conjectured that the
topological space of typical vertices of a finitely forcible graphon is always
compact. We disprove the conjecture by constructing a finitely forcible graphon
such that the associated space is not compact. The construction method gives a
general framework for constructing finitely forcible graphons with non-trivial
properties
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
Fractional colorings of cubic graphs with large girth
We show that every (sub)cubic n-vertex graph with sufficiently large girth
has fractional chromatic number at most 2.2978 which implies that it contains
an independent set of size at least 0.4352n. Our bound on the independence
number is valid to random cubic graphs as well as it improves existing lower
bounds on the maximum cut in cubic graphs with large girth
Fractional coloring of triangle-free planar graphs
We prove that every planar triangle-free graph on vertices has fractional
chromatic number at most
Common graphs with arbitrary chromatic number
Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a
sufficiently large complete graph contains a monochromatic copy of H. In 1962,
Erdos conjectured that the random 2-edge-coloring minimizes the number of
monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta
to all graphs. In the late 1980s, the conjectures were disproved by Thomason
and Sidorenko, respectively. A classification of graphs whose number of
monochromatic copies is minimized by the random 2-edge-coloring, which are
referred to as common graphs, remains a challenging open problem. If
Sidorenko's Conjecture, one of the most significant open problems in extremal
graph theory, is true, then every 2-chromatic graph is common, and in fact, no
2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While
examples of 3-chromatic common graphs were known for a long time, the existence
of a 4-chromatic common graph was open until 2012, and no common graph with a
larger chromatic number is known.
We construct connected k-chromatic common graphs for every k. This answers a
question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab.
Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov
[London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This
also answers in a stronger form the question raised by Jagger, Stovicek and
Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common
graph with chromatic number at least four.Comment: Updated to include reference to arXiv:2207.0942
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