For every finite multiplicative group , let () denote the smallest integer so that every sequence of elements of has a subsequence of length (length ) which has product equal to 1 in some order.
Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.
A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .
A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .
Conjecture Let be an integer and a -regular graph. If is a class 1 graph, then .
Setup Fix a tree and for every vertex a non-negative integer which we think of as the amount of gold at .
2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex of the tree, takes the gold at this vertex, and then deletes . The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.
Conjecture Let be an eulerian graph of minimum degree , and let be an eulerian tour of . Then admits a decomposition into cycles none of which contains two consecutive edges of .
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is for every sets 
and 
:
-flow 
on 
is an orientation 
of 
from the edge set of 
, for all 
, and 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
(
) denote the smallest integer 
so that every sequence of 
(length 
) which has product equal to 1 in some order. 
for every finite group 
has a nowhere-zero 
, and it is denoted by 
.
is a class 1 graph if its edge chromatic number is 
be an integer and 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
. 
contains a directed cycle of length at least 
. 
? 
for principal funcoid 
and a set 
of funcoids of appropriate sources and destinations. 
contains 
so that all pairwise distances between points in 
-
(mod 
) for every vertex 
. 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
has equivalence covering number 
. 
and for every vertex 
a non-negative integer 
which we think of as the amount of gold at 
cycles. 
, and let 
be an eulerian tour of 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
are simple finite graphs, then 
. 
is the 
.
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