Conjecture The following statements are equivalent for every endofuncoid and a set : \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .
Conjecture Let be a graph and be a positive integer. The power of , denoted by , is defined on the vertex set , by connecting any two distinct vertices and with distance at most . In other words, . Also subdivision of , denoted by , is constructed by replacing each edge of with a path of length . Note that for , we have . Now we can define the fractional power of a graph as follows: Let be a graph and . The graph is defined by the power of the subdivision of . In other words . Conjecture. Let be a connected graph with and be a positive integer greater than 1. Then for any positive integer , we have . In [1], it was shown that this conjecture is true in some special cases.
The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
Conjecture If two graphs on vertices have the same deck, then they are isomorphic.
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?
Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.
Let be a non-empty finite set. Given a partition of , the stabilizer of , denoted , is the group formed by all permutations of preserving each block of .
Problem () Find a sufficient condition for a sequence of partitions of to be complete, i.e. such that the product of their stabilizers is equal to the whole symmetric group on . In particular, what about completeness of the sequence , given a partition of and a permutation of ?
Conjecture (Beneš) Let be a uniform partition of and be a permutation of such that . Suppose that the set is transitive, for some integer . Then
Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
Conjecture For every fixed graph , there exists a constant , so that every graph without an induced subgraph isomorphic to contains either a clique or an independent set of size .
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 ?
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is a 
for all primes 
, where 
is prime. 
be a 
is connected regarding 
.
and a set 
:
there exists a totally ordered set 
such that 
, 
, and for every partion 
of 
into two sets 
, 
such that 
, we have ![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
. 
be a graph and 
be a positive integer. The 
power of 
, is defined on the vertex set 
, by connecting any two distinct vertices 
and 
with distance at most 
. Also 
, is constructed by replacing each edge 
of 
, we have 
.
. The graph 
is defined by the 
power of the 
subdivision of 
.
and 
be a positive integer greater than 1. Then for any positive integer 
, we have 
.
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
-
(mod 
) for every vertex 
. 
vertices have the same deck, then they are isomorphic. 
so that both 
are prime.
and a function 
such that if 
, then every saturated 
has cardinality at least 
? 
such that every 
-minor-free graph has at most 
cliques? 
arc-disjoint directed cycles. 
for every endo-
? 
edges whose deletion destroys every triangle. 
grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15. 
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
be a non-empty finite set. Given a partition 
of 
, is the group formed by all permutations of 
.
) Find a sufficient condition for a sequence of partitions 
of 
is equal to the whole symmetric group 
on 
, given a partition 
be a uniform partition of 
be a permutation of 
. Suppose that the set 
is transitive, for some integer 
. Then 
, every 
-diregular oriented graph on at most 
vertices has a Hamilton cycle. 
such that for every graph ![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
so that all pairwise distances between points in 
are rational? 
has a homomorphism to 
. 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
, there exists a constant 
, so that every graph 
. 
where 
for 
? 
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