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,
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.
Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?
Conjecture Define a array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array , the sequence is: -> -> -> -> -> -> -> -> -> -> -> , and we now have a fixed point (loop of one array).
The process always results in a loop of 1, 2, or 3 arrays.
Conjecture Is it possible to color edges of the complete graph using colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
Equivalently: is the star chromatic index of linear in ?
Conjecture Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .
Conjecture If is a bridgelesscubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .
Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?
Conjecture Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition It is easy to prove that is the infinitely small right neighborhood filter of point .
If proved true, the conjecture then can be generalized to a wider class of posets.
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.
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 For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .
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 complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
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has chromatic number at most 
.
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 
plus one or two. In other words, ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
contains a directed cycle of length at least 
. 
generators and exponent 
, is it necessarily finite? 
in the hypercube. 
-neighbour bootstrap process in the hypercube. 
such that every 
with specified vertices 
and 
contains an out-branching rooted at 
grid. The first player (if any) to occupy a set of 
array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array ![$ [1; 1] $](/files/tex/73f8649de444361674c157a2fe98e0c5783f1c46.png)
, the sequence is: ![$ [1; 2] $](/files/tex/83c3d9d7589f716ed6b0f05d26d36dabe8ba47aa.png)
-> ![$ [1, 2; 1, 1] $](/files/tex/a6a696aec4e84df6bc046cf6d30a4df80e156a14.png)
-> ![$ [1, 2; 3, 1] $](/files/tex/98b2f3e4134422c8a286f0326fc2f57ca9be2ab7.png)
-> ![$ [1, 2, 3; 2, 1, 1] $](/files/tex/19a9d24510f4551fa45b950aed32efed0636b355.png)
-> ![$ [1, 2, 3; 3, 2, 1] $](/files/tex/b00d573110ba8c2472820409103dd4cbd7bff7cc.png)
-> ![$ [1, 2, 3; 2, 2, 2] $](/files/tex/2aae90b65ae3f116402a1c0143127f80d86acdf0.png)
-> ![$ [1, 2, 3; 1, 4, 1] $](/files/tex/d614399c4078a70fbffb24eb99816a0974423175.png)
-> ![$ [1, 2, 3, 4; 3, 1, 1, 1] $](/files/tex/bf4dd82eaf95bd71a88cb78d3367efa2c7e4c942.png)
-> ![$ [1, 2, 3, 4; 4, 1, 2, 1] $](/files/tex/a4ef5900a9dde9b32311afe7dee2b143f42c405f.png)
-> ![$ [1, 2, 3, 4; 3, 2, 1, 2] $](/files/tex/ff367376ed7bb74fc6207273d7157062b83a1be6.png)
-> ![$ [1, 2, 3, 4; 2, 3, 2, 1] $](/files/tex/81ba2b3608ddba2fc95d32b74b70279f3f5adc5b.png)
, and we now have a fixed point (loop of one array).
-connected planar graph, then 
has a Hamilton cycle. 
paths. 
using 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
, where 
is some ordinal. This 
as 
.
of 
be the complete funcoid corresponding to the usual topology on extended real line ![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
. Let 
be the order on this set. Then 
is a complete funcoid. 
is the infinitely small right neighborhood filter of point ![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
. 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
on 
) 
(The join operation is taken on the lattice of filters with reversed order.) 
and for every vertex 
a non-negative integer 
which we think of as the amount of gold at 
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
-colored countably infinite complete subgraph. Then 
, 
, or 
.
-flow. 
which is slice, is it a ribbon knot?
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
and every rational 
on 
clauses drawn uniformly at random from the set of formulas on 
simultaneously satisfiable clauses. 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
. 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
matrix, then there is an 
of ![$ [A A] $](/files/tex/d1e9d82c656535b507686183e640178057fae455.png)
so that 
is nonzero. 
be a matroid of rank 
let 
be the number of closed sets of rank 
.
is unimodal. 
is prime and 
, we write 
if 
satisfies 
. The problem of finding such an integer 
(with 
) is the Discrete Log Problem.
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
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