272 research outputs found
Twin-width IV: ordered graphs and matrices
We establish a list of characterizations of bounded twin-width for
hereditary, totally ordered binary structures. This has several consequences.
First, it allows us to show that a (hereditary) class of matrices over a finite
alphabet either contains at least matrices of size , or at
most for some constant . This generalizes the celebrated Stanley-Wilf
conjecture/Marcus-Tardos theorem from permutation classes to any matrix class
over a finite alphabet, answers our small conjecture [SODA '21] in the case of
ordered graphs, and with more work, settles a question first asked by Balogh,
Bollob\'as, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes
of ordered graphs. Second, it gives a fixed-parameter approximation algorithm
for twin-width on ordered graphs. Third, it yields a full classification of
fixed-parameter tractable first-order model checking on hereditary classes of
ordered binary structures. Fourth, it provides a model-theoretic
characterization of classes with bounded twin-width.Comment: 53 pages, 18 figure
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
- …