Hypergraph colouring

A proper hypergraph colouring assigns colours to the vertices of a hypergraph so that no hypergraph edge is monochromatic. Each colour class is a hypergraph independent set. This is often called weak colouring, to distinguish it from requiring every pair of vertices in each edge to have different colours.

The chromatic index of a hypergraph is the minimum number of colours needed to colour its edges so that intersecting edges have different colours. Each colour class is a matching in a hypergraph. It is at least the maximum hypergraph vertex degree, and at most $r(\Delta -1)+1$ for an $r$-uniform hypergraph of maximum degree $\Delta$, by greedily colouring edges: an edge meets at most $r(\Delta -1)$ others.

Given an $r$-uniform hypergraph $H$ and $K$ colours, construct an $(r+1)$-uniform auxiliary hypergraph with one task vertex for every $e\in E(H)$ and one resource vertex $(v,c)$ for every original vertex and colour. The assignment edge for $(e,c)$ contains the task $e$ and all resources $(v,c)$ with $v\in e$. A matching in a hypergraph of assignments is exactly a proper partial edge colouring. Task degrees are $K$; resource degrees are $d_H(v)$. Pairwise auxiliary hypergraph codegrees are at most $\max (1,{\max }_{v\ne w}{d}_H(v,w))$. Thus nearly equal original degrees and small original hypergraph codegrees transfer to the setting of a Rödl nibble. A locally balanced Rödl nibble must control the unmatched task count in every original star, not merely the total uncovered proportion.

A hypergraph has Property B if its vertices of a hypergraph can be coloured with two colours so that every hypergraph edge meets both colour classes. An $r$-uniform hypergraph with fewer than $2^{r-1}$ edges has Property B, by a fair independent colouring and the union bound. Conversely, for $r\ge 2$, take $N=2r^2$ vertices and independently sample $m=\lceil (N\log 2+1)2^r\rceil$ uniformly random $r$-subsets. In any fixed colouring an edge is monochromatic with probability at least $2^{-r}$: convexity of binomial coefficients reduces to the balanced colouring, and $\left(\genfrac{}{}{0ex}{}{r^2}{r}\right)/\left(\genfrac{}{}{0ex}{}{2r^2}{r}\right)\ge 2^{-r-1}$. The union bound over $2^N$ colourings gives a probability at most $2^N{e}^{-m{2}^{-r}}<1$ of any proper colouring. Removing repeated edges leaves a hypergraph without Property B and with $O(r^2{2}^r)$ edges.

A three-uniform hypergraph is bipartite when its vertex set has a partition into two parts such that every hyperedge meets both parts. All such triples on parts of sizes $a,b$ number $a\left(\genfrac{}{}{0ex}{}{b}{2}\right)+b\left(\genfrac{}{}{0ex}{}{a}{2}\right)=ab(a+b-2)/2$. For fixed total order this is maximized by balanced parts.

The hypergraph chromatic number is the minimum number of colours in a proper hypergraph colouring. If $H$ has $N$ vertices and maximum hypergraph independent set size $a$, then $\chi (H)\ge N/a$, since the colour classes partition its vertex set into hypergraph independent sets.