Conjecture If in a bridgeless cubic graph the cycles of any -factor are odd, then , where denotes the oddness of the graph , that is, the minimum number of odd cycles in a -factor of .
If , are graphs, a function is called cycle-continuous if the pre-image of every element of the (binary) cycle space of is a member of the cycle space of .
Problem Does there exist an infinite set of graphs so that there is no cycle continuous mapping between and whenever ?
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
Conjecture Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.
Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
Problem Is there a minimum integer such that the vertices of any digraph with minimum outdegree can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least ?
Problem Let and be two -uniform hypergraph on the same vertex set . Does there always exist a partition of into classes such that for both , at least hyperedges of meet each of the classes ?
Conjecture For every rational and every rational , there is no polynomial-time algorithm for the following problem.
Given is a 3SAT (3CNF) formula on variables, for some , and clauses drawn uniformly at random from the set of formulas on variables. Return with probability at least 0.5 (over the instances) that is typical without returning typical for any instance with at least simultaneously satisfiable clauses.
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.
Conjecture Let be a graph and be a positive integer. The power of , denoted by , is defined on the vertex set , by connecting any two distinct vertices and with distance at most . In other words, . Also subdivision of , denoted by , is constructed by replacing each edge of with a path of length . Note that for , we have . Now we can define the fractional power of a graph as follows: Let be a graph and . The graph is defined by the power of the subdivision of . In other words . Conjecture. Let be a connected graph with and be a positive integer greater than 1. Then for any positive integer , we have . In [1], it was shown that this conjecture is true in some special cases.
Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.
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the cycles of any 
-factor are odd, then 
, where 
denotes the oddness of the graph 
-regular directed graph with no parallel arcs, then 
arc-disjoint directed cycles. 
are graphs, a function 
is called cycle-continuous if the pre-image of every element of the (binary) cycle space of 
so that there is no cycle continuous mapping between 
and 
whenever 
? 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
then 
for every funcoid 
and atomic f.o. 
and 
on the source and destination of 
then 
for every funcoid 
, 
. 
or 
which divides both 
(see 
(see 
. 
be a circuit in a bridgeless cubic graph 
denote the pebbling number of a graph 
. 
such that the vertices of any digraph with minimum outdegree 
can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least 
and 
be two 
-uniform hypergraph on the same vertex set 
. Does there always exist a partition of 
such that for both 
, at least 
hyperedges of 
meet each of the classes 
. 
and every rational 
on 
clauses drawn uniformly at random from the set of formulas on 
simultaneously satisfiable clauses. 
power of 
, is defined on the vertex set 
, by connecting any two distinct vertices 
and 
with distance at most 
. Also 
, is constructed by replacing each edge 
of 
, we have 
.
. The graph 
is defined by the 
power of the 
subdivision of 
.
and 
be a positive integer greater than 1. Then for any positive integer 
, we have 
.
containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 
around the north and south poles. 
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