Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
The crossing number of is the minimum number of crossings in all drawings of in the plane.
The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.
Conjecture The following statements are equivalent for every endofuncoid and a set : \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
Conjecture For every , there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .
A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Problem Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.