The crossing number of is the minimum number of crossings in all drawings of in the plane.
The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.
Conjecture Let is a family of multifuncoids such that each is of the form where is an index set for every and is a set for every . Let every for some multifuncoid of the form regarding the filtrator . Let is a graph-composition of (regarding some partition and external set ). Then there exist a multifuncoid of the form such that regarding the filtrator .
Conjecture For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.
Problem Is there a minimum integer such that the vertices of any digraph with minimum outdegree can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least ?
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.
Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
Conjecture Every arrangement graph of a set of great circles is -colourable.
Conjecture Let be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists such that is an element of at least half the members of .
Conjecture A total coloring of a graph is an assignment of colors to the vertices and the edges of such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph , , equals the minimum number of colors needed in a total coloring of . It is an old conjecture of Behzad that for every graph , the total chromatic number equals the maximum degree of a vertex in , plus one or two. In other words,
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?
Problem Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .
Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .
There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .
Conjecture For which values of and are there bi-colored graphs on vertices and different colors with the property that all the monochromatic colorings have unit weight, and every other coloring cancels out?
Conjecture Is it possible to color edges of the complete graph using colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
Equivalently: is the star chromatic index of linear in ?
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of 
is the minimum number of crossings in all drawings of 
-dimensional (hyper)cube 
is the graph whose vertices are all binary sequences of length 
is a family of multifuncoids such that each 
is of the form 
where 
is an index set for every 
and 
is a set for every 
. Let every 
for some multifuncoid 
of the form 
regarding the filtrator 
. Let 
is a graph-composition of 
). Then there exist a multifuncoid 
of the form 
such that 
regarding the filtrator 
. 
edge-disjoint triangles, then there is a set of 
edges whose deletion destroys every triangle. 
then 
for every funcoid 
and atomic f.o. 
and 
on the source and destination of 
then 
for every funcoid 
, 
. 
and fixed colouring 
of 
with 
colours, there exists 
such that every colouring of the edges of 
contains either 
vertices whose edges are coloured with at most 
colours. 
be abelian groups and let 
and 
satisfy 
and 
. If there is a 
to 
, then every graph with a B-flow has a B'-flow. 
.
of the projective plane is octahedral if all lines in 
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
arc-disjoint directed cycles. 
such that the vertices of any digraph with minimum outdegree 
be a set of 
positive integers and set ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
are simple finite graphs, then 
. 
is the 
has an acyclic induced subdigraph of order at least 
. 
. 
has a complete graph on 
vertices as a minor. 
points in the plane in general position contains a subset of 
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. 
such that 
is an assignment of colors to the vertices and the edges of 
, equals the minimum number of colors needed in a total coloring of 
plus one or two. In other words, ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
in 
, let the symmetry group of 
ie: isotopy classes of diffeomorphisms of 
of 
is the unlink. Let 
be the subgroup of 
consisting of diffeomorphisms of 
given by restricting the diffeomorphism to the 
is a signed symmetric group -- the wreath product of a symmetric group with 
. 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
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