577 research outputs found
On Saturated -Sperner Systems
Given a set , a collection is said to
be -Sperner if it does not contain a chain of length under set
inclusion and it is saturated if it is maximal with respect to this property.
Gerbner et al. conjectured that, if is sufficiently large with respect to
, then the minimum size of a saturated -Sperner system
is . We disprove this conjecture
by showing that there exists such that for every and there exists a saturated -Sperner system
with cardinality at most
.
A collection is said to be an
oversaturated -Sperner system if, for every
, contains more
chains of length than . Gerbner et al. proved that, if
, then the smallest such collection contains between and
elements. We show that if ,
then the lower bound is best possible, up to a polynomial factor.Comment: 17 page
Saturation in the Hypercube and Bootstrap Percolation
Let denote the hypercube of dimension . Given , a spanning
subgraph of is said to be -saturated if it does not
contain as a subgraph but adding any edge of
creates a copy of in . Answering a question of Johnson and Pinto, we
show that for every fixed the minimum number of edges in a
-saturated graph is .
We also study weak saturation, which is a form of bootstrap percolation. A
spanning subgraph of is said to be weakly -saturated if the
edges of can be added to one at a time so that each
added edge creates a new copy of . Answering another question of Johnson
and Pinto, we determine the minimum number of edges in a weakly
-saturated graph for all . More generally, we
determine the minimum number of edges in a subgraph of the -dimensional grid
which is weakly saturated with respect to `axis aligned' copies of a
smaller grid . We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and
Computin
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond
The \emph{choice number} of a graph , denoted , is the minimum
integer such that for any assignment of lists of size to the vertices
of , there is a proper colouring of such that every vertex is mapped to
a colour in its list. For general graphs, the choice number is not bounded
above by a function of the chromatic number.
In this thesis, we prove a conjecture of Ohba which asserts that
whenever . We also prove a
strengthening of Ohba's Conjecture which is best possible for graphs on at most
vertices, and pose several conjectures related to our work.Comment: Master's Thesis, McGill Universit
Beyond Ohba's Conjecture: A bound on the choice number of -chromatic graphs with vertices
Let denote the choice number of a graph (also called "list
chromatic number" or "choosability" of ). Noel, Reed, and Wu proved the
conjecture of Ohba that when . We
extend this to a general upper bound: . Our result is sharp for
using Ohba's examples, and it improves the best-known
upper bound for .Comment: 14 page
A Dichotomy Theorem for Circular Colouring Reconfiguration
The "reconfiguration problem" for circular colourings asks, given two
-colourings and of a graph , is it possible to transform
into by changing the colour of one vertex at a time such that every
intermediate mapping is a -colouring? We show that this problem can be
solved in polynomial time for and is PSPACE-complete for
. This generalizes a known dichotomy theorem for reconfiguring
classical graph colourings.Comment: 22 pages, 5 figure
Finitely forcible graphons with an almost arbitrary structure
Graphons are analytic objects representing convergent sequences of large
graphs. A graphon is said to be finitely forcible if it is determined by
finitely many subgraph densities, i.e., if the asymptotic structure of graphs
represented by such a graphon depends only on finitely many density
constraints. Such graphons appear in various scenarios, particularly in
extremal combinatorics.
Lovasz and Szegedy conjectured that all finitely forcible graphons possess a
simple structure. This was disproved in a strong sense by Cooper, Kral and
Martins, who showed that any graphon is a subgraphon of a finitely forcible
graphon. We strenghten this result by showing for every that
any graphon spans a proportion of a finitely forcible graphon
Finitely forcible graphons with an almost arbitrary structure
Graphons are analytic objects representing convergent sequences of large graphs. A graphon is said to be finitely forcible if it is determined by finitely many subgraph densities, i.e., if the asymptotic structure of graphs represented by such a graphon depends only on finitely many density constraints. Such graphons appear in various scenarios, particularly in extremal combinatorics.
Lovasz and Szegedy conjectured that all finitely forcible graphons possess a simple structure. This was disproved in a strong sense by Cooper, Kral and Martins, who showed that any graphon is a subgraphon of a finitely forcible graphon. We strenghten this result by showing for every ε>0 that any graphon spans a 1−ε proportion of a finitely forcible graphon
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