Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
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 ?
A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Problem Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.
Suppose is a finite group, and is a positive integer dividing . Suppose that has exactly solutions to . Does it follow that these solutions form a subgroup of ?
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?
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Conjecture Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
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 .
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is a countable connected graph then its third 
to be the maximum degree, 
to be the size of the largest 
to be the chromatic number of 
for every graph 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
Square free? 
-norm of a path partition on a directed graph 
is no more than the maximal size of an induced 
, that every sufficiently large 
-
vertices whose deletion results in a 
and 
-neighbour bootstrap process in the hypercube. 
. The graph 
is defined to be the 
-power of the 
.
. Then 
. 
? 
of a graph 
? 
covering design, or covering, is a family of 
-set, such that each 
-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by 
.
. Find a procedure for constructing minimal coverings. 
. Suppose that 
. Does it follow that these solutions form a subgroup 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? 
, let 
be the smallest integer 
such that the symmetric group 
on the set of all words of length 
(
times), where 
is the shuffle permutation defined by 
, and 
is the exchange group consisting of all permutations in 
letters in the words. 
, for all 
, where 
is some ordinal. This 
as 
.
in the 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
, there is an integer 
so that every strongly 
are matroids on 
and 
for every partition 
of 
with 
which is independent in every 
. 
denote the graph with vertex set consisting of all lines through the origin in 
and two vertices adjacent in 
? 
-flow 
on 
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 
. 
.
, the union of any 
-degenerate graph. 
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