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?
Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?
Conjecture Let be the disjoint union of the graphs and and let be a surface. Is it true that every optimal drawing of on has the property that and are disjoint?
Conjecture It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?
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let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
, it is impossible to cover a square of side greater than 
with 
unit squares. 
be a prime natural number. Find all primes 
, such that 
.
contains 
such that 
meets 
in at most one vertex, 
. 
. 
are matroids on 
and 
for every partition 
of 
with 
which is independent in every 
. 
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
, the 
, is defined as the sum of all 
by using partitions. 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
. 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
is defined to be the mimic number of any positive integer 
. 
be a graph. If 
are two integers, a 
-colouring of 
is a function 
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 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
be the unique integer (with respect to a fixed 
) such that
is a prime iff ![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
contains every oriented tree of order 
and 
and let 
be a surface. Is it true that every optimal drawing of 
has a homomorphism to 
. 
people admits a solution is 
. 
does there exists a 
such that every graph with average degree smaller than 
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