Conjecture   There is an absolute constant such that for every hexagonal graph and vertex weighting ,

Conjecture   Let be a planar graph of maximum degree . The chromatic number of its square is

• at most if ,
• at most if ,
• at most if .

Conjecture   There is a constant such that the list chromatic number of any bipartite graph of maximum degree is at most .

Conjecture   Every prism over a -connected cubic planar graph can be decomposed into two Hamilton cycles.

Conjecture   Every simple -uniform hypergraph on vertices which contains no complete -uniform hypergraph on four vertices has at most hyperedges.

Conjecture   Every simple -uniform hypergraph on vertices which contains no complete -uniform hypergraph on five vertices has at most hyperedges.

Conjecture   Every -connected graph with a Hamilton cycle has a second Hamilton cycle.

Conjecture   If is a -connected planar graph, then has a Hamilton cycle.

Conjecture   Every digraph in which each vertex has outdegree at least contains directed cycles such that meets in at most one vertex, .

Conjecture   For every , there is an integer so that every strongly -connected tournament has edge-disjoint Hamilton cycles.

Conjecture   For , every -diregular oriented graph on at most vertices has a Hamilton cycle.

Question   Are there any switching-nonreconstructible digraphs on twelve or more vertices?

Conjecture   Every simple graph on five or more vertices is switching-reconstructible.

Conjecture   Every 4-connected toroidal graph has a Hamilton cycle.

Conjecture   Every planar graph without 4-cycles and 5-cycles is 3-colourable.

Conjecture   Every planar graph is acyclically 5-choosable.

Problem   What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?

Conjecture   If has at most edge-disjoint triangles, then there is a set of edges whose deletion destroys every triangle.

Conjecture   If is a simple triangle-free graph, then there is a set of at most edges whose deletion destroys every odd cycle.

Problem   Let and be two -uniform hypergraph on the same vertex set . Does there always exist a partition of into classes such that for both , at least hyperedges of meet each of the classes ?

Problem  Let be a fixed positive real number and a fixed graph. Is it NP-hard to determine whether a graph on vertices and minimum degree contains and -factor?