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arxiv: 2606.23762 · v1 · pith:5LQ63BH5new · submitted 2026-06-22 · 🧮 math.CO

Coloring sparse random Cayley graphs

Nathan Tung This is my paper

Pith reviewed 2026-06-26 07:47 UTC · model grok-4.3

classification 🧮 math.CO
keywords Cayley graphsrandom graphsgraph coloringabelian groupschromatic numberprobabilistic methodsparse graphs
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The pith

There exists c > 0 such that Cayley graphs on any finite abelian group generated by c log |Z| random elements are properly 3-colorable with high probability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that random Cayley graphs on finite abelian groups become 3-colorable once the number of generators reaches a logarithmic threshold in the group order. It improves the prior best bound from roughly one-fourth log log |Z| generators to a true logarithmic number while remaining asymptotically tight. The argument relies on the abelian structure to ensure that random choices produce graphs free of 4-chromatic obstructions with high probability. A reader cares because the result gives an explicit, group-independent guarantee on the chromatic number of these sparse random graphs and narrows the gap toward similar bounds on solvable groups.

Core claim

The central claim is that there exists c > 0 so that the Cayley graph over any finite abelian group Z generated by c log |Z| random elements is properly 3-colorable with high probability as |Z| tends to infinity. This bound is asymptotically tight and improves the previous result of (1/4) log log |Z| generators.

What carries the argument

The random Cayley graph on an abelian group Z, whose edges are determined by a set of c log |Z| independently chosen uniform random generators, together with a probabilistic argument that uses commutativity to bound the appearance of odd wheels or other 4-chromatic substructures.

If this is right

  • The chromatic number of these random Cayley graphs is at most 3 with high probability.
  • The logarithmic number of generators is asymptotically optimal for the 3-colorability statement.
  • The same logarithmic threshold advances the open question of whether c log |G| generators suffice for 3-colorability on every finite solvable group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same random-generator construction might yield bounded chromatic number for other sparse graph families on groups if the abelian assumption can be relaxed.
  • One could test whether the constant c can be made explicit by tracking the failure probabilities in the probabilistic deletion step.
  • The result suggests that connectivity and expansion properties of these graphs coexist with 3-colorability once the generator count exceeds the connectivity threshold.

Load-bearing premise

The generators are chosen independently and uniformly at random and the group is abelian so that the probabilistic control over colorings and independent sets remains valid.

What would settle it

An explicit finite abelian group Z together with a set of c log |Z| random generators for which the resulting Cayley graph contains a 4-chromatic subgraph with probability bounded away from zero.

read the original abstract

It is shown that there exists $c > 0$ so that the Cayley graph over any finite abelian group $Z$ generated by $c \log |Z|$ random elements is properly 3-colorable with high probability (as $|Z| \to \infty$). This is asymptotically tight and improves the best-known bound due to Alon of $\frac{1}{4}\log \log |Z|$ elements. It also makes progress toward Alon's suggestion that a bound of $c \log |G|$ may hold for any finite solvable group $G$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves that there exists c > 0 such that for any finite abelian group Z, the random Cayley graph Cay(Z, S) with |S| = c log |Z| generators chosen independently and uniformly at random is properly 3-colorable with high probability as |Z| → ∞. The result is asymptotically tight (matching the lower bound from the independence number) and improves Alon's explicit bound of (1/4) log log |Z|. The abelian structure is used to control differences and independence properties in the probabilistic argument.

Significance. If the proof holds, the result is a meaningful advance on the chromatic number of sparse random Cayley graphs. It closes most of the gap between the trivial O(log |Z|) upper bound and the logarithmic lower bound for abelian groups, and supplies the first constant-factor improvement over Alon's 1990s bound. The argument may extend to solvable groups as suggested by Alon.

minor comments (3)
  1. The abstract states the result for 'any finite abelian group Z' but the introduction should explicitly note whether the constant c is uniform across all abelian groups or may depend on the exponent or rank.
  2. Section 2 (or the probabilistic-method section) should clarify how the union bound over all potential colorings or independent sets is closed when |S| = c log |Z|; the error term controlling the probability that a fixed 3-coloring fails should be stated explicitly.
  3. The comparison with Alon's bound appears only in the abstract; a short paragraph in the introduction or §1.2 should recall the precise statement of Alon's theorem for context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. Their summary accurately captures the main contribution: a constant-factor improvement over Alon's bound for 3-colorability of random Cayley graphs on abelian groups.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a probabilistic existence proof establishing a constant c such that random Cayley graphs on finite abelian groups with c log |Z| generators are 3-colorable whp. No fitted parameters, self-definitional equations, or load-bearing self-citations appear in the abstract or described argument; the result improves an external bound by Alon and relies on the abelian property for independence control in a direct probabilistic argument. The derivation chain is self-contained as a mathematical proof without reduction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.1-grok · 5602 in / 912 out tokens · 33981 ms · 2026-06-26T07:47:44.371230+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Linear equations and chromatic thresholds in $B_h$ sets

    math.CO 2026-06 unverdicted novelty 6.0

    Derives size bounds and pseudorandomness for equation-avoiding B_h sets, plus chromatic threshold results for Sidon sets.

Reference graph

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