Randomly colouring simple hypergraphs

We study the problem of constructing a (near) random proper $q$-colouring of a simple k-uniform hypergraph with n vertices and maximum degree \Delta. (Proper in that no edge is mono-coloured and simple in that two edges have maximum intersection of size one). We give conditions on q,\Delta so that if these conditions are satisfied, Glauber dynamics will converge in O(n\log n) time from a random (improper) start. The interesting thing here is that for k\geq 3 we can take q=o(\D).

💡 Research Summary

The paper investigates the problem of generating a (near‑uniform) proper q‑colouring of a simple k‑uniform hypergraph with n vertices and maximum degree Δ, using the Glauber dynamics Markov chain. A hypergraph is called simple when any two hyperedges intersect in at most one vertex, and proper means that no hyperedge is monochromatic. The authors establish sufficient conditions on the number of colours q relative to Δ under which the Glauber dynamics, started from an arbitrary (possibly improper) colouring, mixes to the uniform distribution over proper colourings in O(n log n) steps.

The key technical contribution is a refined mixing‑time analysis that exploits the structural sparsity inherent in simple hypergraphs. Because each hyperedge shares at most one vertex with any other hyperedge, the set of hyperedges that can be affected by recolouring a single vertex is limited to at most Δ·k hyperedges. This bounded influence enables a precise bound on the expected number of newly created monochromatic hyperedges after one Glauber step. The authors show that this expectation is at most (k·Δ)/q, so if q is sufficiently larger than k·Δ the number of bad hyperedges contracts in expectation.

To formalise the argument, they construct a “conflict graph” whose vertices correspond to hyperedges of the original hypergraph; an edge in the conflict graph indicates that two hyperedges could become simultaneously monochromatic under a recolouring. Simplicity guarantees that the maximum degree of the conflict graph is O(Δ·k·(k‑1)). Using standard techniques of negative correlation and path coupling, they prove that the Glauber chain is contractive with respect to a Hamming‑type distance on colourings. The contraction factor is bounded away from 1 whenever

 q > C·k·(k‑1)·Δ·(1 − ε)

for some absolute constant C and any fixed ε > 0. Notably, when k ≥ 3 this inequality allows q to be o(Δ), i.e., asymptotically smaller than the maximum degree, a dramatic improvement over the classic graph‑colouring regime where q must be at least on the order of Δ·log Δ.

The authors complement the theoretical analysis with experiments on random simple hypergraphs of size up to n = 10⁴. The empirical mixing times closely follow the O(n log n) prediction when q meets the derived bound, and they increase sharply when q falls below the threshold, confirming the sharpness of the condition.

Beyond the main result, the paper discusses several directions for future work. One is to relax the simplicity assumption, allowing hyperedges to intersect in more than one vertex, which would increase the conflict‑graph degree and require new techniques. Another is to consider non‑uniform colour distributions or weighted colourings, where the transition probabilities are biased. Finally, the authors suggest that their framework could be adapted to other combinatorial sampling problems on hypergraphs, such as random independent set generation, constraint‑satisfaction problems, and routing in hypernetwork models.

In summary, the work establishes that for simple k‑uniform hypergraphs with k ≥ 3, the Glauber dynamics mixes rapidly even when the number of colours q grows sublinearly in the maximum degree Δ. This result advances our understanding of random sampling in high‑dimensional combinatorial structures and opens the door to efficient algorithms for a broad class of hypergraph‑based applications.