Conjecture   Does a perfect cuboid exist?

Conjecture   Every graph with minimum degree at least 7 contains a -minor.

Conjecture   Every 7-connected graph contains a -minor.

Conjecture   (solved) If then for every funcoid and atomic f.o. and on the source and destination of correspondingly.

Conjecture   If then for every funcoid and , .

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 .

Conjecture   There is no odd perfect number.

Question   For every does there exists a such that every graph with average degree smaller than and girth at least has a matching-cut?

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   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.

Problem   Characterize the -graphs.

Conjecture   If is a bridgeless cubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .

Conjecture   There exist infinitely many positive integers so that both and are prime.

Question   Does every strongly regular graph have either itself or a complete graph as a core?

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 four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.

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?

Question   Does there exists, for infinitely many integers , a convex polygon on vertices whose extension complexity is ?

Conjecture   for some filter objects , . Particularly, is this formula true for ?

Conjecture   for some filter objects , .

Conjecture   Every 3-connected cubic planar bipartite graph is Hamiltonian.

Conjecture   For any integer , it is impossible to cover a square of side greater than with unit squares.

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.