Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.
Conjecture Let be an -uniform-partite hypergraph. If is the maximum number of pairwise disjoint edges in , and is the size of the smallest set of vertices which meets every edge, then .
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
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?
The star chromatic index of a graph is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.
Question Is it true that for every (sub)cubic graph , we have ?
Problem Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?
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.
login/create account
has chromatic number at most 
.
be an 
-
is the maximum number of pairwise disjoint edges in 
is the size of the smallest set of vertices which meets every edge, then 
. 
be a simple graph with 
vertices and list chromatic number 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
denote the 
satisfies 
. 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
. 
with 
so that 
has a 
. 
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
-connected planar graph, then 
has a Hamilton cycle. 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
edges whose deletion destroys every odd cycle. 
of a graph 
? 
edges. Define a form parameter for a polyhedron as 
where 
is the number of vertices. What is the distribution of 
for 
? 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
and 
it holds
if 
if 
; \item 
; \item 
. 
with specified vertices 
and 
for reloid 
for every funcoid 
(see below mentioned online article).
Drupal
CSI of Charles University