Triangle-Free Coloring in LOCAL via Resilient Lovász Local Lemma

Abstract: The Lovász Local Lemma (LLL) is a probabilistic tool that has been shown to be of central importance in the study of distributed algorithms. For example, the constructive LLL is known to be complete for the class of locally-checkable labeling problems with $o(\log n)$ randomized complexities in the LOCAL model. One classic application of the LLL is in coloring graphs with some sparse structure, such as triangle-free graphs. Triangle-free coloring therefore serves as a benchmark problem for techniques for sublogarithmic randomized distributed algorithms. The state-of-the-art distributed triangle-free coloring algorithm of Pettie and Su [ICALP 2013, Information and Computation 2015] uses $\fracΔ{k}$ colors (where $k$ can be up to $(\frac14 - \varepsilon)\ln Δ$) and consists of $O(k+\log^* n)$ applications of the distributed LLL. However, the distributed LLL is itself a difficult problem; despite significant study, the fastest algorithms known require $O(\log_Δn)$ or $O(\fracΔ{\logΔ})+\log^{O(1)}\log n$ rounds. In this work, we adapt the Pettie-Su's algorithm so that the resulting LLL instances can be solved in $\log^{O(1)}\log n$ rounds, by employing the 'resilience' definition of Davies [SODA 2023]. This gives an $O(k)+ \log^{O(1)}\log n$ complexity (since the LLL is not needed when $k= \log^{ω(1)}\log n$), essentially causing the LLL steps to no longer be the bottleneck of the algorithm. As a corollary we obtain the first $\log^{O(1)}\log n$-round algorithms for coloring triangle-free graphs with $o(Δ)$ colors. The same framework also yields a companion girth-$5$ algorithm, using $(1+\varepsilon)Δ/\ln Δ$ colors in $O(k)+ \log^{O(1)}\log n$ rounds, matching the best known existential upper bound for the number of colors.

• The paper develops a resilient Lovász Local Lemma framework that colors triangle-free graphs with Δ/k colors in O(k) + log^{O(1)} log n rounds, for k ≤ (1/4 − ε) ln Δ, with high probability.
• The algorithm constructs color-balanced r-partitions that control both neighborhood and per-color dependencies, enabling efficient resilience analysis without union bounds over exponentially many resampling patterns.
• The results give the first o(Δ)-coloring of triangle-free graphs in polyloglog rounds and achieve (1+o(1))Δ/ln Δ colors for girth-5 graphs, while leaving the optimal triangle-free constant as an open problem.

Overview and motivation

This paper addresses a long-standing bottleneck in distributed graph coloring: the cost of the Lovász Local Lemma (LLL) subroutine inside iterative coloring algorithms for sparse graphs. The starting point is the framework of Pettie and Su (2608.13357), which colors any triangle-free graph of maximum degree $\Delta$ with $\Delta/k$ colors, for $k \le (\tfrac14 - \varepsilon)\ln\Delta$, in $O(k + \log^* n)$ rounds when $\Delta$ is large relative to $n$. For smaller $\Delta$, however, each phase of that algorithm must be certified by an application of the distributed LLL, and the best general-purpose distributed LLL algorithms require either $O(\log n)$ rounds or $O(\Delta/\log\Delta) + \log^{O(1)}\log n$ rounds. Consequently, in the regime $\Delta = (\log n)^{\Theta(1)}$ — precisely where the LLL is hardest — the LLL calls dominate the round complexity.

The contribution is to restructure the Pettie–Su algorithm so that its LLL instances fall within the resilient partition framework of Davies (SODA 2023), which solves such instances in $\Delta/k$0 rounds for an $\Delta/k$1-partition. This yields an $\Delta/k$2-round algorithm — a nearly exponential improvement over prior bounds in the difficult regime — while retaining the same asymptotic color count.

Main results

The paper establishes two theorems plus a corollary:

• Triangle-free graphs: for any fixed $\Delta/k$3, any $\Delta/k$4, and any $\Delta/k$5, there is a randomized LOCAL algorithm that colors every triangle-free graph with $\Delta/k$6 colors in $\Delta/k$7 rounds with high probability.
• Corollary: triangle-free graphs can be colored with $\Delta/k$8 colors in $\Delta/k$9 rounds. This is the first sub-$k \le (\tfrac14 - \varepsilon)\ln\Delta$0-color result at this complexity, and it is polynomially close to tight given the $k \le (\tfrac14 - \varepsilon)\ln\Delta$1 lower bound of Brandt et al., which already applies to coloring constant-degree trees.
• Girth-5 graphs: $k \le (\tfrac14 - \varepsilon)\ln\Delta$2 colors in $k \le (\tfrac14 - \varepsilon)\ln\Delta$3 rounds for $k \le (\tfrac14 - \varepsilon)\ln\Delta$4, matching the best known existential bound (Kim's result) for this class.

Both results extend to list-coloring, since the analysis is phrased entirely in terms of evolving palette sizes. The color counts match the best achievable by any nontrivial distributed algorithm; notably, Molloy's existential improvement of the triangle-free constant from $k \le (\tfrac14 - \varepsilon)\ln\Delta$5 to $k \le (\tfrac14 - \varepsilon)\ln\Delta$6 remains out of reach distributively, and the authors flag this as open.

Technical approach

The core idea, stated informally by the authors, is that LLL instances whose bad events assert that certain quantities deviate too far from expectation — where those quantities are governed primarily by one-hop neighborhoods and admit $k \le (\tfrac14 - \varepsilon)\ln\Delta$7 additive slack — should be $k \le (\tfrac14 - \varepsilon)\ln\Delta$8-resilient and hence solvable in $k \le (\tfrac14 - \varepsilon)\ln\Delta$9 rounds. In coloring applications, slack in concentration bounds translates directly into a negligible number of extra colors, so resilience comes "for free" from the palette budget.

The key new combinatorial object is a color-balanced $O(k + \log^* n)$0-partition (Definition 4): a partition of the active vertices into $O(k + \log^* n)$1 parts such that every vertex has at most $O(k + \log^* n)$2 neighbors and at most $O(k + \log^* n)$3 $O(k + \log^* n)$4-neighbors (neighbors whose palette contains color $O(k + \log^* n)$5) in each part. The second condition is essential because the bad events of the Pettie–Su process are per-color: they concern palette sizes, average $O(k + \log^* n)$6-degrees, and residual degrees, so controlling only total degree within parts is insufficient. The paper proves that such partitions exist whp and can be constructed in $O(k + \log^* n)$7 rounds via an auxiliary 1-resilient LLL instance in which vertices choose parts uniformly at random.

The main algorithmic difficulty is verifying resilience of the three bad-event families ($O(k + \log^* n)$8 for palette size, $O(k + \log^* n)$9 for average $\Delta$0-degree, $\Delta$1 for residual degree) under adversarial resampling of arbitrary subsets of a single part. Two obstacles stand out. First, the events depend on two-hop neighborhoods, so resampling propagates through intermediate vertices; the triangle-free condition ensures that the relevant variable sets decompose disjointly, but individual $\Delta$2-degrees do not decrease smoothly, forcing the analysis to track average $\Delta$3-degree rather than per-color bounds. Second, the resilience definition quantifies over exponentially many subsets $\Delta$4 of a part, making naive union bounds impossible. The proofs circumvent this by identifying "symptom" events (e.g., $\Delta$5, $\Delta$6, $\Delta$7) that are uniform over all $\Delta$8: once these first-sample events are excluded, the conditional probability of failure under any resampling pattern is bounded by $\Delta$9 or better. A signature-map device assigns witness colors to certify which resampled selections could newly uncolor a vertex, keeping the accounting per-color and independent.

The parameter schedule follows Variant A of Pettie–Su with modified constants: sampling probability $n$0 with $n$1, slack parameter $n$2, floor $n$3, and stopping threshold $n$4. Once residual degree drops below $n$5, the remaining graph is colored via the shattering-based $n$6-list-coloring algorithm of Chang–Li–Pettie combined with Ghaffari–Grunau network decomposition, in $n$7 rounds using $n$8 fresh colors — negligible against the $n$9 budget.

For girth-5 graphs, local independence is strong enough to maintain direct per-color $\Delta$0-degree bounds instead of averages, recovering Kim's constant-1 regime with a shorter argument. The genuinely new proof component there is resilience of the per-color event $\Delta$1, handled via three carefully separated effects (direct contribution of the resampled block, palette loss through witnesses, and induced uncoloring), each bounded independently thanks to the absence of 4-cycles.

Significance for the distributed LLL

Beyond coloring, the paper widens the class of LLL instances known to be solvable in polyloglog rounds. The instances solved here are described as among the most complex yet handled efficiently, and they are the first special cases shown directly to fit the $\Delta$2-resilience framework with $\Delta$3; previously, superconstant resilience was used only as an intermediate step toward the general LLL result. The authors' informal claim — that concentration-type bad events with neighborhood-local influence and additive slack are prime candidates for fast resilient solving — suggests applicability beyond coloring, e.g., to problems requiring label frequencies within thresholds, which constitute most concrete distributed LLL applications.

Limitations and open questions

Several caveats qualify the results. The analysis assumes $\Delta$4 is a fixed positive constant and $\Delta$5 sufficiently large; the authors state that extending to $\Delta$6 slowly should be possible by tracking dependencies explicitly but do not carry this out. The round bound $\Delta$7 hides constants in the polyloglog exponent, and the improvement is specific to the regime $\Delta$8; for larger $\Delta$9 the original Pettie–Su analysis already suffices. The color count $O(\log n)$0 with $O(\log n)$1 still leaves a factor-of-4 gap to the existential optimum $O(\log n)$2 for triangle-free graphs, and no distributed algorithm matching Molloy's bound is known. More broadly, the general distributed LLL retains an exponential gap: no $O(\log n)$3-round worst-case algorithm exists, and whether one does — or whether the $O(\log n)$4 lower bound can be strengthened — remains open.

Conclusion

The paper converts the LLL bottleneck of the Pettie–Su triangle-free coloring framework into a non-bottleneck by proving that its per-round LLL instances are resilient with respect to color-balanced partitions, yielding the first $O(\log n)$5-round algorithms for coloring triangle-free graphs with $O(\log n)$6 colors and girth-5 graphs with $O(\log n)$7 colors. The technical machinery — color-balanced partitions, symptom-event arguments avoiding union bounds over resampling subsets, and signature-based certification — constitutes a reusable template for solving concentration-flavored LLL instances efficiently, and sharpens the boundary between tractable and general LLL instances in the LOCAL model.