List Total Colouring Conjecture ★★
Author(s): Borodin; Kostochka; Woodall
Conjecture If is the total graph of a multigraph, then .
Keywords: list coloring; Total coloring; total graphs
Partitioning the Projective Plane ★★
Author(s): Noel
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.
Keywords: Partitioning; projective plane
Kriesell's Conjecture ★★
Author(s): Kriesell
Conjecture Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
Keywords: Disjoint paths; edge-connectivity; spanning trees
2-colouring a graph without a monochromatic maximum clique ★★
Conjecture If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.
Keywords: maximum clique; Partitioning