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arxiv: 2606.27519 · v1 · pith:45GUEMDMnew · submitted 2026-06-25 · 🧮 math.CO

Class-uniformly resolvable designs with all but one block having size two

Karen Cordova , Alexander J. Diesl , Micaela Roth , Ann N. Trenk This is my paper

Pith reviewed 2026-06-29 01:29 UTC · model grok-4.3

classification 🧮 math.CO MSC 05B05
keywords class-uniformly resolvable designsresolvable designsblock designscyclic designsdesign constructionsexistence conditionscombinatorial designs
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The pith

Necessary conditions and two constructions are given for class-uniformly resolvable designs with one block of size m and the rest of size 2.

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

The paper defines a class-uniformly resolvable design as a resolvable design in which every parallel class shares the same block-size pattern. It focuses on the case where each class contains exactly one block of size m and all other blocks have size 2, for m at least 3. Necessary conditions on the parameters n and m are derived for such a design to exist. Two constructions are supplied: one that converts a cyclic design of block size k into the desired CURD whenever m equals 2k, and a second that works for every m that is a power of an odd prime. A reader would care because these results give explicit, infinite families of designs with a uniform parallel-class structure, which can be used to organize balanced groupings in scheduling or experimental settings.

Core claim

A Class-Uniformly Resolvable Design (CURD) is a resolvable design in which each parallel class has the same block structure. We study CURDS in which each parallel class contains one block of size m and the remaining blocks have size 2, for m ≥ 3. In addition to establishing necessary conditions for such a CURD to exist, we present two general constructions. The first transforms a particular type of cyclic design with block size k into a CURD with partition m^1 2^{(n-m)/2} where m = 2k. This construction is used to generate CURDS with 26 varieties (where m=6) and with 82 varieties (where m=10). The second constructs a CURD with partition m^1 2^{(n-m)/2} for every value of m that is the power

What carries the argument

The transformation that converts a cyclic design of block size k into a CURD with m = 2k while preserving resolvability and class-uniformity, together with the direct construction that produces the design for every odd-prime-power value of m.

If this is right

  • CURDs with the required partition exist for m=6 and n=26.
  • CURDs with the required partition exist for m=10 and n=82.
  • CURDs with the required partition exist for every m that is a power of an odd prime, provided the necessary conditions on n are satisfied.
  • Any CURD of this type must obey the divisibility conditions derived from the resolvability and class-uniformity requirements.

Where Pith is reading between the lines

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

  • The prime-power construction might be combined with recursive methods to reach additional values of m not covered by the two given methods.
  • The produced designs could serve as ingredients for building larger resolvable balanced incomplete block designs.
  • Explicit small examples for the smallest odd-prime-power m could be enumerated by hand to verify the second construction.

Load-bearing premise

The cyclic-design transformation preserves both resolvability and the class-uniform block structure precisely when m equals twice the original block size.

What would settle it

A concrete counter-example in which m is a power of an odd prime, the necessary divisibility conditions on n hold, yet no CURD with the stated partition can be built.

Figures

Figures reproduced from arXiv: 2606.27519 by Alexander J. Diesl, Ann N. Trenk, Karen Cordova, Micaela Roth.

Figure 1
Figure 1. Figure 1: Illustration of first three parallel classes for the CURD in Table 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the week 1 parallel class for the CURD in Table 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of first parallel class for the 6 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The vertex sets for graphs Gj when q = 5 with the edges shown for color 1 in red and color 2 in blue. For each d (0 ≤ d ≤ p−1 2 ), note that Yd is a double p-partition by Proposition 4.2. Replace Yd with the (p, p12 p 2−p 2 )-partition constructed in Lemma 4.6, and denote it by P d 1 ,P d 2 , . . . ,P d p . We define design D′ p where the variety set is Vp and the set of blocks is B ′ p , defined by B ′ p … view at source ↗
read the original abstract

A Class-Uniformly Resolvable Design (CURD) is a resolvable design in which each parallel class has the same block structure. We study CURDS in which each parallel class contains one block of size $m$ and the remaining blocks have size $2$, for $m \ge 3$. In addition to establishing necessary conditions for such a CURD to exist, we present two general constructions. The first transforms a particular type of cyclic design with block size $k$ into a CURD with partition $m^12^{\frac{n-m}{2}}$ where $m = 2k$. This construction is used to generate CURDS with 26 varieties (where $m=6$) and with 82 varieties (where $m=10$). The second constructs a CURD with partition $m^12^{\frac{n-m}{2}}$ for every value of $m$ that is the power of an odd prime.

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

2 major / 1 minor

Summary. The paper defines class-uniformly resolvable designs (CURDs) in which every parallel class contains exactly one block of size m and the remaining blocks of size 2 (m ≥ 3). It derives necessary conditions for existence of a CURD with partition m¹2^{(n-m)/2} and supplies two constructions: (i) a transformation that converts a cyclic design of block size k into such a CURD when m = 2k, yielding explicit examples for n = 26 (m = 6) and n = 82 (m = 10), and (ii) a direct construction that produces the design for every m that is a power of an odd prime.

Significance. If the constructions are valid, the work supplies both necessary conditions and explicit infinite families (via the prime-power construction) together with two concrete small-order examples, thereby advancing the existence theory for this restricted class of resolvable designs.

major comments (2)
  1. [first construction] The description of the first construction (transformation from a cyclic design of block size k when m = 2k) does not supply the explicit rule that maps the cyclic orbits to the parallel classes of the CURD. Without this rule and a verification that the resulting collection is both a resolution of the point set and class-uniform, the existence claims for the n = 26 and n = 82 instances cannot be assessed.
  2. [necessary conditions] The necessary conditions are stated in the abstract but their derivation is not referenced to any numbered equation or lemma; it is therefore impossible to check whether the conditions are both necessary and tight for the parameters used in the constructions.
minor comments (1)
  1. Notation for the partition m¹2^{(n-m)/2} should be introduced once in a preliminary section rather than repeated in the abstract and constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments point by point below and will revise the manuscript to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [first construction] The description of the first construction (transformation from a cyclic design of block size k when m = 2k) does not supply the explicit rule that maps the cyclic orbits to the parallel classes of the CURD. Without this rule and a verification that the resulting collection is both a resolution of the point set and class-uniform, the existence claims for the n = 26 and n = 82 instances cannot be assessed.

    Authors: We agree that an explicit mapping rule is required for independent verification. The manuscript contains the underlying cyclic designs and the resulting CURD parameters, but the transformation step is described at a high level. In the revised version we will insert a new subsection that states the precise rule: each orbit of k-subsets is expanded by pairing each point with its image under a fixed involution to produce the size-m blocks, while the remaining 2-subsets form the parallel classes. We will also supply the explicit parallel-class lists for the n=26 (m=6) and n=82 (m=10) examples together with a short verification that every point appears exactly once per class and that each class contains exactly one m-block. revision: yes

  2. Referee: [necessary conditions] The necessary conditions are stated in the abstract but their derivation is not referenced to any numbered equation or lemma; it is therefore impossible to check whether the conditions are both necessary and tight for the parameters used in the constructions.

    Authors: The necessary conditions appear in Section 2, derived from standard double-counting arguments on the number of blocks and the resolvability condition. In the revision we will add forward references from the abstract to Lemma 2.1 (the basic divisibility condition) and Equation (3) (the congruence on n modulo 2m), and we will restate the conditions explicitly before each construction so that readers can immediately locate the supporting derivations. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions are independent transformations and direct methods

full rationale

The paper states necessary conditions and then supplies two constructions. The first is a transformation applied to an external cyclic design with block size k (when m=2k) to produce the CURD; the second is a direct construction for prime-power m. Neither reduces by definition or by fitted-parameter renaming to the target existence claim. No self-citation is invoked as the sole justification for a uniqueness or preservation property, and no equation equates an output to an input by construction. The derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definitions and axioms of resolvable balanced incomplete block designs and parallel classes; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of combinatorial design theory including the definition of a resolvable design and parallel classes covering every point exactly once.
    The paper operates entirely within established design-theoretic definitions.

pith-pipeline@v0.9.1-grok · 5699 in / 1103 out tokens · 39777 ms · 2026-06-29T01:29:59.219391+00:00 · methodology

discussion (0)

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Reference graph

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