[Paper Review] On $r$-colorability of random hypergraphs

Андрей Борисович Купавский, D. A. Shabanov|arXiv (Cornell University)|Oct 6, 2011

Constraint Satisfaction and Optimization15 references3 citations

TL;DR

This paper establishes a new threshold for r-colorability in random k-uniform hypergraphs H(n,k,p), proving that when r^{k-1} grows sufficiently fast relative to n and k, the hypergraph is asymptotically almost surely r-colorable under a specific edge probability p. The result improves prior bounds by leveraging a refined random recoloring method and a carefully chosen function φ(k) = Θ(√(ln ln k / ln k)).

ABSTRACT

The work deals with the threshold probablity for r-colorability in the binomial model H(n,k,p) of a random k-uniform hypergraph. We prove a lower bound for this threshold which improves the previously known results in the wide range of the parameters r=r(n) and k=k(n).

Motivation & Objective

• To determine the threshold probability p* for r-colorability in the binomial random hypergraph model H(n,k,p).
• To close the gap between known upper and lower bounds for r-colorability when r and k grow with n.
• To extend previous results on 2-colorability to general r-colorability in sparse random hypergraphs.
• To improve the asymptotic threshold for r-colorability by introducing a function φ(k) that captures the logarithmic dependence on k.

Proposed method

• Uses the random recoloring method to analyze the existence of proper r-colorings in sparse random hypergraphs.
• Applies a probabilistic method based on constructing random colorings with controlled monochromatic edge probabilities.
• Introduces a function φ(k) = Θ(√(ln ln k / ln k)) to refine the threshold condition and improve the bound.
• Imposes conditions on r and k such that r^{k-1} ≥ 6 ln n and r^{k-1} ≤ n^{(1−δ)/2} for δ ∈ (0,1), ensuring the threshold applies.
• Derives a sufficient condition on edge probability: p ≤ (1/2) · r^{k−1} / (k^{1+φ(k)}) · n / (n choose k).
• Adapts techniques from prior work on chromatic number to the list coloring setting, extending results to r-choosability.

Experimental results

Research questions

• RQ1 What is the threshold probability p* for r-colorability in the random hypergraph model H(n,k,p) when r and k grow with n?
• RQ2 How can the random recoloring method be refined to improve the lower bound on the r-colorability threshold?
• RQ3 What role does the function φ(k) = Θ(√(ln ln k / ln k)) play in tightening the threshold condition?
• RQ4 Can the results on 2-colorability be generalized to r-colorability for r ≥ 2 in the sparse regime?
• RQ5 Under what conditions on r and k is H(n,k,p) asymptotically almost surely r-colorable?

Key findings

• For r^{k−1} ≥ 6 ln n and r^{k−1} ≤ n^{(1−δ)/2} with δ ∈ (0,1), the hypergraph H(n,k,p) is r-colorable with high probability when p ≤ (1/2) · r^{k−1} / (k^{1+φ(k)}) · n / (n choose k).
• The threshold condition improves upon previous results by Alon and Spencer (1995) and Achlioptas and Moore (2006) in the wide parameter range where r and k grow with n.
• The function φ(k) = 4⌊√(ln k / ln(2 ln k))⌋^{−1} is used to refine the exponent in the denominator, enhancing the bound.
• The result holds asymptotically as n → ∞, with the probability of r-colorability approaching 1 under the stated conditions.
• The method extends to r-choosability, showing that the same threshold condition ensures r-choosability with high probability.
• The bound is tighter than earlier results for r ≥ 3 and sufficiently large k, particularly in the regime r^{k−1} ≥ n^{(1−δ)/2}.