8 research outputs found
On the Erd\H{o}s-Tuza-Valtr Conjecture
The Erd\H{o}s-Szekeres conjecture states that any set of more than
points in the plane with no three on a line contains the vertices of a convex
-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that
any set of more than points in a
plane either contains the vertices of a convex -gon, points lying on a
concave downward curve, or points lying on a concave upward curve. They
also showed that the generalization is actually equivalent to the
Erd\H{o}s-Szekeres conjecture.
We prove the first new case of the Erd\H{o}s-Tuza-Valtr conjecture since the
original 1935 paper of Erd\H{o}s and Szekeres. Namely, we show that any set of
points in the plane with no three points on a line and no
two points sharing the same -coordinate either contains 4 points lying on a
concave downward curve or the vertices of a convex -gon.Comment: 16 pages, 8 figure
Twin-width VIII: delineation and win-wins
We introduce the notion of delineation. A graph class is said
delineated if for every hereditary closure of a subclass of
, it holds that has bounded twin-width if and only if
is monadically dependent. An effective strengthening of
delineation for a class implies that tractable FO model checking
on is perfectly understood: On hereditary closures of
subclasses of , FO model checking is fixed-parameter tractable
(FPT) exactly when has bounded twin-width. Ordered graphs
[BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively
delineated, while subcubic graphs are not. On the one hand, we prove that
interval graphs, and even, rooted directed path graphs are delineated. On the
other hand, we show that segment graphs, directed path graphs, and visibility
graphs of simple polygons are not delineated. In an effort to draw the
delineation frontier between interval graphs (that are delineated) and
axis-parallel two-lengthed segment graphs (that are not), we investigate the
twin-width of restricted segment intersection classes. It was known that
(triangle-free) pure axis-parallel unit segment graphs have unbounded
twin-width [BGKTW, SODA '21]. We show that -free segment graphs, and
axis-parallel -free unit segment graphs have bounded twin-width, where
is the half-graph or ladder of height . In contrast, axis-parallel
-free two-lengthed segment graphs have unbounded twin-width. Our new
results, combined with the known FPT algorithm for FO model checking on graphs
given with -sequences, lead to win-win arguments. For instance, we derive
FPT algorithms for -Ladder on visibility graphs of 1.5D terrains, and
-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure
Randomized Communication and Implicit Graph Representations
We study constant-cost randomized communication problems and relate them to implicit graph representations in structural graph theory. Specifically, constant-cost communication problems correspond to hereditary graph families that admit constant-size adjacency sketches, or equivalently constant-size probabilistic universal graphs (PUGs), and these graph families are a subset of families that admit adjacency labeling schemes of size O(log n), which are the subject of the well-studied implicit graph question (IGQ). We initiate the study of the hereditary graph families that admit constant-size PUGs, with the two (equivalent) goals of (1) understanding randomized constant-cost communication problems, and (2) understanding a probabilistic version of the IGQ. For each family studied in this paper (including the monogenic bipartite families, product graphs, interval and permutation graphs, families of bounded twin-width, and others), it holds that the subfamilies are either stable (in a sense relating to model theory), in which case they admit constant-size PUGs, or they are not stable, in which case they do not. The correspondence between communication problems and hereditary graph families allows for a new method of constructing adjacency labeling schemes. By this method, we show that the induced subgraphs of any Cartesian products are positive examples to the IGQ. We prove that this probabilistic construction cannot be derandomized by using an Equality oracle, i.e. the Equality oracle cannot simulate the k-Hamming Distance communication protocol. We also obtain constant-size sketches for deciding for vertices , in any stable graph family with bounded twin-width. This generalizes to constant-size sketches for deciding first-order formulas over the same graphs
Extremal and probabilistic results for regular graphs
In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness.
We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory.
We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph
of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs.
Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum