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.
Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.
Given integers , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).
Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).
Let be the smallest integer such that the graph is rearrangeable.
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.
Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is
Problem What is the best upper bound on circular choosability for planar graphs?
Conjecture A total coloring of a graph is an assignment of colors to the vertices and the edges of such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph , , equals the minimum number of colors needed in a total coloring of . It is an old conjecture of Behzad that for every graph , the total chromatic number equals the maximum degree of a vertex in , plus one or two. In other words,
Conjecture For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .
Conjecture Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
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.
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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.) 
exist a 
. 
such that for every graph 
, ![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
with 
so that 
has a 
. 
has a homomorphism to 
. 
-connected finite graph, is there an assignment of lengths 
to the edges of 
-geodesic cycle is 
does there exists a 
such that every graph with average degree smaller than 
. The representation function 
for 
is given by the rule 
. We call 
is never 
.
has chromatic number at most 
. 
, the 2-stage Shuffle-Exchange graph/network, denoted 
, is the simple 
-regular bipartite graph with the ordered pair 
of linearly labeled parts 
and 
, where 
, such that vertices 
and 
are adjacent if and only if 
(see Fig.1).
, the 
-stage Shuffle-Exchange graph/network, denoted 
, is the proper (i.e., respecting all the orders) concatenation of 
identical copies of 
be the smallest integer 
such that the graph 
. 
is the maximal degree of a graph 
-colorable. 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
be a graph. If 
and 
are two integers, a 
-colouring of 
from 
to 
such that 
for each edge 
. Given a list assignment 
of 
a set of non-negative integers, an 
such that 
for every 
. A list assignment 
and 
for each vertex 
. Given such a list assignment 
, the graph 
is an assignment of colors to the vertices and the edges of 
, equals the minimum number of colors needed in a total coloring of 
plus one or two. In other words, ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
of 
points in the plane, not all collinear, there is a point in 
lines determined by 
cycles. 
such that for any pair 
there are 
edge-disjoint paths from 
to 
. 
into 
so that the graph induced by 
is acyclic for 
. 
, 
so that 
does not contain an odd edge-cut. 
a non-negative integer 
which we think of as the amount of gold at 
, is defined as the sum of all 
by using partitions. 
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