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.
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 \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.
Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
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 ?
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.
Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
Conjecture Every arrangement graph of a set of great circles is -colourable.
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be a circuit in a bridgeless cubic graph 
. Then there is a five cycle double cover of 
, there exist positive integers 
, 
, 
such that 
.
be a set of 
positive integers and set ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
, the 
-th 
, is defined as the sum of all 
by using partitions. 
for reloid 
. 
for every funcoid 
(see below mentioned online article).
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 
and every positive integer 
so that every simple 
-regular graph 
has a 
with 
. 
for 
. 
of a locally finite graph 
is a sequence of elements from a multiplicative group of order 
so that 
. 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
(The join operation is taken on the lattice of filters with reversed order.) 
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 
? 
contains every 2-regular graph on 
vertices as a minor. 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
the following inequality holds 
rectangular 
inside an 
? 
(along the track) away from every other runner. 
. 
of great circles on a sphere with no three circles meeting at a point. The arrangement graph of 
-colourable. 
denote ''
divisible by 
, the mimic function, 
, is given by,
is defined to be the mimic number of any positive integer 
. 
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