Colourings without monochromatic disjoint pairs

On Colourings of Hypergraphs Without Monochromatic Fano Planes

For k-uniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual extremal or Turán function, i.e., the maximum number of hyperedges of an n-vertex k-unifor...

On monochromatic component size for improper colourings

This paper concerns improper λ-colourings of graphs and focuses on the sizes of the monochromatic components (i.e., components of the subgraphs induced by the colour classes). Consider the following three simple operations, which should, heuristically, help reduce monochromatic component size: (a) assign to a vertex the colour that is least popular among its neighbours; (b) change the colours o...

Monochromatic Cycle Partitions in Local Edge Colourings

An edge colouring of a graph is said to be an r-local colouring if the edges incident to any vertex are coloured with at most r colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any 2-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number r, we show that the vertex set o...

Disjoint NP-Pairs

We study the question of whether the class DisNP of disjoint pairs (A,B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provi...

Representable Disjoint NP-Pairs