Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.
Conjecture If is the maximal degree of a graph , then is strongly -colorable.
Conjecture Let be a cubic graph with no bridge. Then there is a coloring of the edges of using the edges of the Petersen graph so that any three mutually adjacent edges of map to three mutually adjancent edges in the Petersen graph.
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 .
Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is
Problem What is the best upper bound on circular choosability for planar graphs?
A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .
A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .
Conjecture Let be an integer and a -regular graph. If is a class 1 graph, then .
Problem Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?
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and 
, the graph 
has equivalence covering number 
. 
be a locally compact topological group. If 
be a positive integer. We say that a graph 
is the maximal degree of a graph 
-colorable. 
has a complete graph on 
vertices as a minor. 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
such that 
meets 
in at most one vertex, 
. 
then 
for every funcoid 
and atomic f.o. 
and 
on the source and destination of 
then 
for every funcoid 
, 
. 
is prime and 
, we write 
if 
satisfies 
. The problem of finding such an integer 
(with 
) is the Discrete Log Problem.
be a graph. If 
are two integers, a 
-colouring of 
from 
to 
such that 
for each edge 
. Given a list assignment 
of 
a set of non-negative integers, an 
such that 
for every 
. A list assignment 
-
and 
for each vertex 
. Given such a list assignment 
, the graph 
are simple finite graphs, then 
. 
is the 
.
for principal funcoid 
of funcoids of appropriate sources and destinations. 
structurally stable diffeomorphism is hyperbolic. 
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
on 
of 
from the edge set of 
, for all 
, and 
. The circular flow number of 
has a nowhere-zero 
, and it is denoted by 
.
be an integer and 
-regular graph. If 
. 
of 
. 
be natural numbers with 
. It follows from the pigeon-hole principle that there exist distinct subsets 
with 
. Is it possible to find such a pair 
in polynomial time? 
is a 
for all primes 
, where 
is prime. 
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