The star chromatic index of a graph is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.
Question Is it true that for every (sub)cubic graph , we have ?
Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.
The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
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.
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
For a finite (additive) abelian group , the Davenport constant of , denoted , is the smallest integer so that every sequence of elements of with length has a nontrivial subsequence which sums to zero.
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has distinct subset sums if distinct subsets of 
have distinct sums.
so that 
whenever 
has distinct subset sums. 
is prime and 
, we write 
if 
satisfies 
. The problem of finding such an integer 
for a given 
(with 
) is the Discrete Log Problem.
of a graph 
is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.
? 
contains a directed cycle of length at least 
. 
be the flow polynomial of a graph 
, the value 
equals the number of 
for every 2-edge-connected graph 
. Suppose X is an m-by-n integer matrix 
. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
. In other words, simplify this formula. 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
. Is it true that for large enough~
? 
, whether there exists a homomorphism from 
for some constant 
be a 
-dimensional smooth submanifold of 
, 
. By the Jordan-Brouwer separation theorem, 
-manifolds which share 
? ie: is 
and a function 
such that if 
, then every saturated 
has cardinality at least 
? 
be a positive integer. We say that a graph 
is the maximal degree of a graph 
-colorable. 
and 
. Then for any neighborhood 
there is 
such that 
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 
, is the smallest integer 
so that every sequence of elements of 
has a nontrivial subsequence which sums to zero.
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
be the unique integer (with respect to a fixed 
) such that
is a prime iff ![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
and 
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