23 research outputs found
Subgraph densities in signed graphons and the local Sidorenko conjecture
We prove inequalities between the densities of various bipartite subgraphs in
signed graphs and graphons. One of the main inequalities is that the density of
any bipartite graph with girth r cannot exceed the density of the r-cycle. This
study is motivated by Sidorenko's conjecture, which states that the density of
a bipartite graph F with m edges in any graph G is at least the m-th power of
the edge density of G. Another way of stating this is that the graph G with
given edge density minimizing the number of copies of F is, asymptotically, a
random graph. We prove that this is true locally, i.e., for graphs G that are
"close" to a random graph.Comment: 20 page
An approximate version of Sidorenko's conjecture
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H
is a bipartite graph, then the random graph with edge density p has in
expectation asymptotically the minimum number of copies of H over all graphs of
the same order and edge density. This conjecture also has an equivalent
analytic form and has connections to a broad range of topics, such as matrix
theory, Markov chains, graph limits, and quasirandomness. Here we prove the
conjecture if H has a vertex complete to the other part, and deduce an
approximate version of the conjecture for all H. Furthermore, for a large class
of bipartite graphs, we prove a stronger stability result which answers a
question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page
Cut distance identifying graphon parameters over weak* limits
The theory of graphons comes with the so-called cut norm and the derived cut
distance. The cut norm is finer than the weak* topology. Dole\v{z}al and
Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin.
Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a
cut distance accumulation graphon can be pinpointed in the set of weak*
accumulation points as a minimizer of the entropy. Motivated by this, we study
graphon parameters with the property that their minimizers or maximizers
identify cut distance accumulation points over the set of weak* accumulation
points. We call such parameters cut distance identifying. Of particular
importance are cut distance identifying parameters coming from subgraph
densities, t(H,*). This concept is closely related to the emerging field of
graph norms, and the notions of the step Sidorenko property and the step
forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step
Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory
Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if
and only if it is step Sidorenko, and that if a graph is norming then it is
step forcing. Further, we study convexity properties of cut distance
identifying graphon parameters, and find a way to identify cut distance limits
using spectra of graphons. We also show that continuous cut distance
identifying graphon parameters have the "pumping property", and thus can be
used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming
graphs, and a new section 3.2 on index pumping in the regularity lemm
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
Common Pairs of Graphs
A graph is said to be common if the number of monochromatic labelled
copies of in a red/blue edge colouring of a large complete graph is
asymptotically minimized by a random colouring with an equal proportion of each
colour. We extend this notion to an asymmetric setting. That is, we define a
pair of graphs to be -common if a particular linear
combination of the density of in red and in blue is asymptotically
minimized by a random colouring in which each edge is coloured red with
probability and blue with probability . We extend many of the results
on common graphs to this asymmetric setting. In addition, we obtain several
novel results for common pairs of graphs with no natural analogue in the
symmetric setting. We also obtain new examples of common graphs in the
classical sense and propose several open problems.Comment: 50 page
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Sidorenko Hypergraphs and Random Tur\'an Numbers
Let denote the maximum number of edges in an
-free subgraph of the random -uniform hypergraph . Building on
recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on
whenever is not Sidorenko. This connection
between Sidorenko's conjecture and random Tur\'an problems gives new lower
bounds on whenever is not Sidorenko, and further
allows us to bound how "far" from Sidorenko an -graph is whenever upper
bounds for are known.Comment: 13 pages (+1 page Appendix), 1 figur