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.

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   Given and , the graph has equivalence covering number .

Question   What is the complexity of the following problem?

Given , determine whether or not

Conjecture   Let be an abelian group of odd order and let satisfy . Then the elements of and may be ordered and so that the sums are pairwise distinct.

Conjecture   Every 4-edge-connected graph has a nowhere-zero 3-flow.

Problem   Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?

Conjecture   There exists a fixed constant so that whenever has distinct subset sums.