Conjecture Suppose that is a -edge-critical graph. Suppose that for each edge of , there is a list of colors. Then is -edge-colorable unless all lists are equal to each other.
Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.
Conjecture If is the maximal degree of a graph , then is strongly -colorable.
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.
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
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 Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.
In particular, this implies:
Conjecture Twin Prime Conjecture: There are an infinite number of twin primes.
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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are independent random variables with ![$ \mathbb{E}[X_i] \leq \mu $](/files/tex/e0268221532981debea25e9446c8ee6f112e1881.png)
, then ![$$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$](/files/tex/03dc1130142ee6fefcc33888e2fb6137211bf327.png)
is a 
-edge-critical graph. Suppose that for each edge 
of 
of 
-edge-colorable unless all lists are equal to each other. 
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
be a positive integer. We say that a graph 
-colorable. 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
and graph 
. 
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 
.
. Find a procedure for constructing minimal coverings. 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
and 
has distinct subset sums if distinct subsets of 
have distinct sums.
so that 
whenever 
has distinct subset sums. 
there is 
such that, for every large 
, there are 
and 
. 
. 
or 
which divides both 
(see 
(see 
and 
and let 
be a surface. Is it true that every optimal drawing of 
such that 
meets 
in at most one vertex, 
. 
for every 
with odd size.
for every 
does there exists a 
for reloid 
for every funcoid 
(see below mentioned online article).
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
. 
using 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
be an elliptic curve over a number field 
. Then the order of the zeros of its 
, at 
is the Mordell-Weil rank of 
. 
-manifold has the homotopy type of the 
, is it diffeomorphic to 
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