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.
Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?
Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.
Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?
2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?
3. Finally, what could one say about higher dimensional analogs of this question?
Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n
Conjecture Suppose runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least (along the track) away from every other runner.
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.
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 .
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?
Conjecture Let be a sequence of points in with the property that for every , the points are distinct, lie on a unique sphere, and further, is the center of this sphere. If this sequence is periodic, must its period be ?
Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .
Conjecture [2] Let be a graph with list chromatic number and . Then
Conjecture Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
An alternating walk in a digraph is a walk so that the vertex is either the head of both and or the tail of both and for every . A digraph is universal if for every pair of edges , there is an alternating walk containing both and
Question Does there exist a locally finite highly arc transitive digraph which is universal?
Conjecture It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?
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, 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 
letters in the words. 
, for all 
such that the list chromatic number of any bipartite graph 
of maximum degree 
is at most 
.
denote the 
diagonal 
exists. 
, the 
, is defined as the sum of all 
by using partitions. 
exist a 
(along the track) away from every other runner. 
) 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. 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
has distinct subset sums if distinct subsets of 
have distinct sums.
whenever 
has distinct subset sums. 
is a finite field with at least 4 elements and 
matrix with entries in 
which have no coordinates equal to zero such that 
. 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
-graph if it has a bipartition 
so that every vertex in 
and every vertex in 
has degree 
. 
be a sequence of points in 
with the property that for every 
, the points 
are distinct, lie on a unique sphere, and further, 
is the center of this sphere. If this sequence is periodic, must its period be 
? 
let 
be the maximum number of vertices of 
, where the minimum is taken over all list assignments 
for all 
. 
and 
. Then ![\[\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.\]](/files/tex/47be18e956355dd433b88b66eabf01a9e3ed5f61.png)
such that for every graph ![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
-connected cubic graph on 
vertices admit a partition into 
if 
if 
. Assume we start with some number 
of the current number. Prove that no matter what the initial number is we eventually reach 
. 
, is the graph with vertex set 
and two points adjacent if the distance between them is an odd integer. 
? 
so that the vertex 
is either the head of both 
and 
or the tail of both 
. A digraph is universal if for every pair of edges 
, there is an alternating walk containing both 
and 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to 
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