Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?
Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
Conjecture An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the vertices' trees, the star with vertices has the most endomorphisms, while the path with vertices has the least endomorphisms.
Conjecture Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.
Problem Let be a tournament with edges colored from a set of three colors. Is it true that must have either a rainbow directed cycle of length three or a vertex so that every other vertex can be reached from by a monochromatic (directed) path?
Given a finite family of graphs and an integer , the Turán number of is the largest integer such that there exists a graph on vertices with edges which contains no member of as a subgraph.
Conjecture For every finite family of graphs there exists an such that .
Conjecture For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.
Conjecture Let be an integer. For every integer , there exists an integer such that for every digraph , either has a pairwise-disjoint directed cycles of length at least , or there exists a set of at most vertices such that has no directed cycles of length at least .
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
Problem Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?
1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
Conjecture Can the approximation ratio be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than -hardness?
Conjecture Let be a sequence of points in with the property that for every , the points are distinct, lie on a unique sphere, and further, is the center of this sphere. If this sequence is periodic, must its period be ?
Conjecture If is a simple graph which is the union of pairwise edge-disjoint complete graphs, each of which has vertices, then the chromatic number of is .
login/create account
such that every 
with specified vertices 
and 
contains an out-branching rooted at 
,
. 
vertices' trees, the star with 
such that for any pair 
there are 
edge-disjoint paths from 
. 
is for every sets 
and 
? 
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
of graphs and an integer 
of 
such that there exists a graph on 
such that 
.
such that the list chromatic number of any bipartite graph 
is at most 
.
and fixed colouring 
of 
with 
such that every colouring of the edges of 
contains either 
vertices whose edges are coloured with at most 
colours. 
generators and exponent 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, or there exists a set 
vertices such that 
has no directed cycles of length at least 
paths. 
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
is not an integer. 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
edges whose deletion destroys every odd cycle. 
if 
, \item at most 
if 
, \item at most 
if 
. 
be a sequence of points in 
with the property that for every 
, the points 
are distinct, lie on a unique sphere, and further, 
is the center of this sphere. If this sequence is periodic, must its period be 
? 
Drupal
CSI of Charles University