Conjecture For every , the sequence in consisting of copes of and copies of has the fewest number of distinct subsequence sums over all zero-free sequences from of length .
An -factor in a graph is a set of vertex-disjoint copies of covering all vertices of .
Problem Let be a fixed positive real number and a fixed graph. Is it NP-hard to determine whether a graph on vertices and minimum degree contains and -factor?
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.
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 ?
Problem () Find a sufficient condition for a straight -stage graph to be rearrangeable. In particular, what about a straight uniform graph?
Conjecture () Let be a simple regular ordered -stage graph. Suppose that the graph is externally connected, for some . Then the graph is rearrangeable.
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
Conjecture There is a function such that for every graph ,
For a finite (additive) abelian group , the Davenport constant of , denoted , is the smallest integer so that every sequence of elements of with length has a nontrivial subsequence which sums to zero.
Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .
Conjecture For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.
Conjecture Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition It is easy to prove that is the infinitely small right neighborhood filter of point .
If proved true, the conjecture then can be generalized to a wider class of posets.
Problem Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .
Conjecture If is a simple graph which is the union of pairwise edge-disjoint complete graphs, each of which has vertices, then the chromatic number of is .
Conjecture Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).
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-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
, the sequence in 
consisting of 
copes of 
and 
copies of 
has the fewest number of distinct subsequence sums over all zero-free sequences from 
. 
-factor in a graph 
is a set of vertex-disjoint copies of 
be a fixed positive real number and 
vertices and minimum degree 
contains and 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to 
denote the 
satisfies 
. 
.
of the projective plane is octahedral if all lines in 
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
, so that every graph 
. 
-minor. 
? 
of co-prime positive integers 
for 
. 
. 
be an integer and let 
. Then either 
minor for some 
or 
edges. Define a form parameter for a polyhedron as 
where 
is the number of vertices. What is the distribution of 
for 
? 
) Find a sufficient condition for a straight 
-stage graph to be rearrangeable. In particular, what about a straight uniform graph? 
) Let 
be a simple regular ordered 
-stage graph. Suppose that the graph 
is externally connected, for some 
. Then the graph 
is rearrangeable. 
of 
and a (non-proper) 
such that edges with the same colour do not cross with respect to 
for some edges 
, then 
and 
receive distinct colours. 
is the minimum integer 
be the graph obtained by subdividing each edge of 
such that for every graph 
edges whose deletion destroys every odd cycle. 
. The representation function 
for 
. We call 
is never 
.
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
, is the smallest integer 
has a nontrivial subsequence which sums to zero.
in 
there is an integer 
be the complete funcoid corresponding to the usual topology on extended real line ![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
. Let 
be the order on this set. Then 
is a complete funcoid. 
is the infinitely small right neighborhood filter of point ![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
. 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
so that the power series 
is bounded for all 
. 
is connected. Then 
has a cycle with length at most 
-minor-free graph is 
-list-colourable for some constant 
. 
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