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.
Question Is the binary affine cube the only 3-connected matroid for which equality holds in the bound where is the circumference (i.e. largest circuit size) of ?
Problem Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?
Conjecture If is a finite field with at least 4 elements and is an invertible matrix with entries in , then there are column vectors which have no coordinates equal to zero such that .
Given integers , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).
Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).
Let be the smallest integer such that the graph is rearrangeable.
Conjecture If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.
Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .
Conjecture If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
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) 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. 
and a function 
such that if 
, then every saturated 
-Sperner system 
has cardinality at least 
? 
. 
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
be a 3-regular graph that contains no cycle of length shorter than 
. Is it true that for large enough~
? 
such that every 
-vertex 
-minor-free graph has at most 
cliques? 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
is defined to be the mimic number of any positive integer 
. 
the only 3-connected matroid for which equality holds in the bound 
where 
is the circumference (i.e. largest circuit size) of 
? 
be natural numbers with 
. It follows from the pigeon-hole principle that there exist distinct subsets 
with 
. Is it possible to find such a pair 
in polynomial time? 
, there is an integer 
so that every strongly 
-connected finite graph, is there an assignment of lengths 
to the edges of 
-
(mod 
) for every vertex 
. 
is a finite field with at least 4 elements and 
is an invertible 
matrix with entries in 
which have no coordinates equal to zero such that 
. 
, the 2-stage Shuffle-Exchange graph/network, denoted 
, is the simple 
of linearly labeled parts 
and 
, where 
, such that vertices 
and 
are adjacent if and only if 
(see Fig.1).
, the 
-stage Shuffle-Exchange graph/network, denoted 
, is the proper (i.e., respecting all the orders) concatenation of 
identical copies of 
be the smallest integer 
such that the graph 
. 
of co-prime positive integers 
for 
. 
. In other words, simplify this formula. 
, then there is a 
contains every 2-regular graph on 
vertices as a minor. 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
is a completary multifuncoid of the form 
, then 
is a completary multifuncoid of the form 
. 
is for every sets 
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