Conjecture Let be a cubic graph with no bridge. Then there is a coloring of the edges of using the edges of the Petersen graph so that any three mutually adjacent edges of map to three mutually adjancent edges in the Petersen graph.
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 A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
Conjecture Let and . Then for any neighborhood there is such that is periodic point of
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for
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.
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.
An oriented colouring of an oriented graph is assignment of colours to the vertices such that no two arcs receive ordered pairs of colours and . It is equivalent to a homomorphism of the digraph onto some tournament of order .
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 .
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.
For and positive integers, the (mixed) van der Waerden number is the least positive integer such that every (red-blue)-coloring of admits either a -term red arithmetic progression or an -term blue arithmetic progression.
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can be replaced by a smaller value, we can also write the conjecture as, for any positive 
be a 
of co-prime positive integers 
for 
. ![\[ w(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil. \]](/files/tex/d2391343543ce03d861e6eb2f4985d52e309525d.png)
. 
vertices, then 
for 
. 
of 
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. 
, let 
be the smallest integer 
such that the symmetric group 
on the set of all words of length 
-letter alphabet can be generated as 
(
times), where 
is the shuffle permutation defined by 
, and 
is the exchange group consisting of all permutations in 
letters in the words. 
, for all ![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
be a digraph. If 
, then 
and 
. Then for any neighborhood 
there is 
such that 
is periodic point of 
vector fields . In the case of diffeos this was proved by Charles Pugh for 
. In the case of Flows this has been solved by Sushei Hayahshy for 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
vertices have the same deck, then they are isomorphic. 
then 
for every funcoid 
and atomic f.o. 
and 
on the source and destination of 
then 
for every funcoid 
, 
. 
covering design, or covering, is a family of 
-set, such that each 
-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by 
.
. Find a procedure for constructing minimal coverings. 
of colours to the vertices such that no two arcs receive ordered pairs of colours 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
for every endo-
. 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
.
of the projective plane is octahedral if all lines in 
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
. The representation function 
for 
is given by the rule 
. We call 
is never 
.
positive integers, the (mixed) van der Waerden number 
is the least positive integer ![$ [1,n] $](/files/tex/661d0acc09fbc5b62f49645a71cf7d831a26563f.png)
admits either a 
, 
. 
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