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Computational methods for finding bi-regular cages

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper's exhaustive generator determines all pairwise non-isomorphic ({r,m};g)-cages for 24 triples and shows no ({4,5};7)-graph exists on 66–68 vertices.

desk verdict Solid computational paper that settles exact orders and counts for two dozen small bi-regular cage triples and improves many bounds; the exhaustive lower-bound claim rests on a well-documented search whose pruning correctness at the critical scale is not independently verified, but no concrete flaw is identified. read the letter →

arxiv 2411.17351 v1 pith:Z7OYNDQC submitted 2024-11-26 math.CO cs.DM

Jan Goedgebeur , Jorik Jooken , Tibo Van den Eede This is my paper · ORCID
classification math.COcs.DM MSC 05C0705C3505C8568R1090C35
keywords bi-regularcagescageproblemgirthexhaustivegenerationbacktrackingalgorithmMoorebounddegreesequencesgraphalgorithms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops computational methods to find bi-regular cages, graphs of a fixed girth whose vertices have exactly two allowed degrees. Its main claim is that a new exhaustive backtracking generator, biregGen, determines all pairwise non-isomorphic ({r,m};g)-cages for 24 triples (r,m,g) that were previously unknown. Running this generator at orders 66, 67 and 68 produces no ({4,5};7)-graph, which raises the lower bound n({4,5};7) from 66 to 69. The paper also improves 122 upper bounds on cage orders: 49 via constructions starting from r-regular graphs, and 73 via a generalization of a gluing theorem for (r,g)-graphs. If the exhaustive claims are correct, these are exact new facts about small bi-regular cages, not merely heuristic bounds.

What carries the argument

The central object is the bi-regular Moore tree (BMT), the analogue of the Moore tree for two degrees r and m. The argument's engine is Theorem 2.4, a closed formula for the order of a BMT in terms of the placement of degree-m vertices on its levels; this formula drives the maximal m-placement pruning rule and the distance-between-degree-m-vertices constraint. The exhaustive generator biregGen is a backtracking search that starts from a BMT augmented by isolated vertices and recursively adds edges, using these pruning rules plus an isomorphism check to guarantee it outputs each graph exactly once. For the upper-bound improvements, the constructions are: adding an edge to an r-regular graph (Construction 3.1), deleting non-adjacent edges and attaching a new vertex (Constructions 3.2 and 3.3), and identifying remote vertices of several (r,g)-graphs (Theorem 4.2).

What would settle it

Run an independent exhaustive generator or a SAT-based search for ({4,5};7)-graphs on 66, 67, and 68 vertices; finding even one such graph would immediately falsify the claimed lower bound of 69 and the completeness of biregGen for those orders.

Watch

Extended reading notes

Core claim

The central claim is that the backtracking algorithm biregGen, built on a bi-regular Moore tree and two new pruning rules, exhaustively generates all pairwise non-isomorphic ({r,m};g)-graphs of a given order for the 24 triples listed in Table 1, yielding the complete cage lists and the nonexistence of ({4,5};7)-graphs on 66, 67, or 68 vertices. The paper further claims that the generalized gluing construction (Theorem 4.2) and the regular-graph constructions improve 73 and 49 upper bounds, respectively, for n({r,m};g). The authors state these as proven computational results, subject to the correctness of the implementation and the completeness of the pruning rules.

Load-bearing premise

The exhaustive results rest on the pruning rules never discarding a graph that could be completed to a valid ({r,m};g)-graph, and since the nonexistence for orders 66–68 was only sanity-checked on smaller orders, an over-pruning bug confined to larger searches would invalidate them.

Editorial extensions

If this is right

  • The 24 exhaustive lists are established; for example, the unique ({3,4};10)-cage on 82 vertices is confirmed by biregGen.
  • The lower bound n({4,5};7) ≥ 69 holds: no ({4,5};7)-graph exists with 66, 67, or 68 vertices.
  • 49 upper bounds on n({r,m};g) are improved by constructions from r-regular graphs, using existing censuses as seeds.
  • The generalized gluing theorem improves 73 further upper bounds, bringing the total number of improved bounds to 122.
  • The same generator could settle n({3,8};8), since running it for order 74 would decide between the known bounds 74 and 75.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the completeness of the pruning rules holds at all orders, the same biregGen approach can be applied to other small open cases, though the paper notes the computational cost grows steeply (about 185 CPU days for order 68 alone).
  • The observation that degree-m vertices are never adjacent in any of the 24 cage lists suggests a possible structural theorem: for girth at least 5, no bi-regular cage has adjacent degree-m vertices; checking this on additional triples would be a natural next step.
  • The generalized gluing theorem may yield infinite families of bi-regular graphs with prescribed girth; the paper's approximate algorithms for finding many vertices and edges at pairwise distance could be refined into exact algorithms for larger orders.
  • The lower-bound claim could be independently checked by encoding the search for orders 66–68 as a SAT or constraint-solving problem; a satisfying assignment would refute the completeness claim.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies bi-regular cages, i.e. ({r,m};g)-graphs of minimum order. It presents three methods: (1) an exhaustive backtracking generator called biregGen, which leads to exhaustive lists of cages for 24 triples (r,m,g) and improves the lower bound for n({4,5};7) from 66 to 69; (2) constructions that start from r-regular graphs and improve 49 upper bounds; and (3) a generalization of a gluing theorem of Aguilar, Araujo-Pardo and Berman, yielding 73 further upper-bound improvements. The paper includes pseudocode, a GitHub repository with code and data, and an appendix with sanity checks against geng and multigraph for smaller orders.

Significance. If correct, the paper is a substantial contribution to the computational study of bi-regular cages. The exhaustive lists for 24 triples are new and provide a rich source of data; the improvement of the lower bound for n({4,5};7) from 66 to 69 is a concrete advance; and the 122 improved upper bounds, obtained through systematic constructions, are a valuable resource. The availability of code and the sanity checks against independent generators are strengths that support reproducibility. The main caveat is that the exhaustiveness of the generator at the critical orders is not independently verified, and the key pruning theorem is not proved in the paper.

major comments (2)
  1. [Section 2.1, Theorem 2.4] Theorem 2.4, which gives the order of a bi-regular Moore tree in terms of the numbers of degree-m vertices on each level, is not proved in the paper; the proof is delegated to the third author's Master's thesis [47]. This theorem is the mathematical basis for the maximal m-placement pruning rule of Section 2.3, and therefore for the exhaustiveness of biregGen. The paper should include a complete, self-contained proof or at least a detailed derivation, because the current text gives only Observation 2.3 and a reference to an external thesis.
  2. [Section 2.5.2 and Appendix A] The improved lower bound n({4,5};7) ≥ 69 rests on the nonexistence of ({4,5};7)-graphs on 66, 67, and 68 vertices, as reported by the implementation of Algorithm 2. The sanity checks in Table 4 compare biregGen with geng and multigraph only for orders up to 29 (and for some cases up to order 20 with larger counts). These checks do not exercise the search at the critical orders 66–68, where one run alone took roughly 185 CPU days. An over-pruning bug or an implementation error that only manifests at larger orders would invalidate the lower-bound claim and some of the exhaustive lists. The authors should provide additional validation for these orders (for example, an independent generator, a formal completeness argument for the pruning rules, or a certificate that the search space was fully exhausted) or explicitly state this as a limitation.
minor comments (5)
  1. [Section 2.5.1, first paragraph] The sentence 'the largest ratio n({r,m};g)/|V_m| in the table is 0.231' is incorrect as stated: for ({3,4};5) the ratio n/|V_m| is at least 13/3 ≈ 4.33. The intended ratio is probably |V_m|/n, which for that triple is at most 3/13 ≈ 0.231. Please correct the wording.
  2. [Algorithm 2, line 2] The condition 'if method was called with graph that is isomorphic with G then return' is informal. Please specify how the isomorphism test is performed (e.g., using nauty) and under what circumstances a previously processed graph is recognized, since this is important for the completeness of the enumeration.
  3. [Section 3, Construction 3.1] Construction 3.1 says to add an edge between two different vertices of G_reg. For simple graphs this only works when the chosen vertices are non-adjacent; this condition should be stated explicitly.
  4. [Section 4.1, last paragraph] The two approximate algorithms for computing the maximum number of vertices and edges at a given pairwise distance are not described. Please provide at least a short outline or a reference, so that the reader can assess how conservative the resulting upper bounds might be.
  5. [Throughout, Section 2] The notation 'BMT' and the level sets L_d, L_{d,k} are used extensively; a small diagram or a reference to Figure 1 would help the reader keep track of the root conventions for even versus odd girth.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; exhaustive census and explicit construction bounds are independent of any fitted input.

full rationale

The paper's main claims are obtained by direct generation and construction. The new lower bound n({4,5};7) >= 69 is the result of executing biregGen at orders 66, 67, and 68 and obtaining zero outputs; no fitted parameter is involved and the claim is not a reformulation of any input bound. The pruning rules in Sections 2.2 and 2.3 are supported by mathematical derivations (Observation 2.3 and Theorem 2.4); while the detailed proof of Theorem 2.4 is delegated to the third author's Master's thesis [47], it is a genuine level-counting calculation with stated assumptions that do not include the target nonexistence result, so this is at most an omitted-proof/correctness risk rather than circularity. The 49 upper bounds in Section 3 are built by explicit constructions from independently published lists of regular graphs listed in Table 2, and the 73 bounds in Section 4 follow from a theorem proved in the paper (Theorem 4.2) generalizing an external result by Aguilar, Araujo-Pardo, and Berman; these are not renamed known results or fit-derived numbers. The self-citations to [26], [27], [47], and [48] are auxiliary (algorithmic inspiration, an earlier hypohamiltonian cage, implementation details) and none carries a load-bearing equivalence between premise and conclusion. Appendix A provides independent sanity checks against geng and multigraph on smaller orders; the remaining concern that pruning completeness at n=68 is not independently verified at that scale is a correctness/exhaustiveness risk, not a circular step under the definitions used here.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard extremal graph theory bounds, the correctness of the cited regular graph censuses used as seeds, and the unverified-in-this-paper details of the BMT order formula. No ad hoc fitted constants or invented entities appear.

assumptions (5)
  • domain assumption A ({r,m};g)-graph contains a bi-regular Moore tree of order at most the bi-regular Moore bound M({r,m};g).
    Used in Section 2 to initialize the exhaustive search; this is the standard Moore-bound argument for regular cages extended to the bi-regular case.
  • domain assumption Theorem 2.4 gives the exact order of a bi-regular Moore tree from its m-placement.
    Proof is deferred to Master's thesis [47, Section 3.1]; the maximal m-placement pruning rule in Section 2.3 depends on it.
  • domain assumption The distance-based lower bounds in Theorem 2.5 from [7] and [3] are correct and applicable to all generated graphs.
    Used as a pruning rule in Section 2.2; the paper cites these results without reproving them.
  • domain assumption The external censuses of r-regular graphs listed in Table 2 are correct and contain the claimed graphs.
    Section 3 uses these lists as seeds; if a listed graph were invalid, the corresponding upper-bound improvement could be invalid.
  • domain assumption The record-holder (r,g)-graphs used in Theorem 4.2 have girth at least g, and the computed numbers s of well-separated vertices and edges are correct or conservative.
    Section 4.1 uses exact algorithms up to order 728 and approximate algorithms beyond; conservative s still gives valid upper bounds.

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Pith. "Pith review of Computational methods for finding bi-regular cages." pith.science (2026). https://pith.science/paper/Z7OYNDQC

@misc{pith2026241117351,
  author       = {Pith},
  title        = {Pith review of: Computational methods for finding bi-regular cages},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z7OYNDQC}},
  note         = {Machine review of arXiv:2411.17351}
}
abstract

An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\geq3$ with vertices of degrees $r$ and $m$ where $2 \leq r < m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.

Figures

Figures reproduced from arXiv: 2411.17351 by the authors.

Figure 1
Figure 1. A BMT T pvwq 3,4,3 . In case only one vertex in the root of the BMT is of degree m, then the BMT corresponds to the tree associated with the bi-regular Moore bound (Eq. (2)) where d “ t, w “ tuw, vwu for g “ 2t and d “ t ´ 1, w “ vw for g “ 2t ´ 1. In this subsection we present a lower bound on the amount of vertices of an ptr, mu; gq-graph given the amount of vertices of degree m on each level of the bi-regular Moo… view at source ↗
Figure 2
Figure 2. One of the pt3, 7u; 5q-cages, which is also hypohamiltonian [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. One of the pt3, 4u; 6q-cages (left) and one of the pt4, 5u; 5q-cages (right). 2.5.2 Improved lower bounds The exhaustive generation algorithm allows us to improve lower bounds as well. That is because if the algorithm does not produce any graphs for the order n equal to the lower bound nlow, then we know that the lower bound is at least nlow ` 1. We were able to improve the lower bound of npt4, 5u; 7q from 66 to 69,… view at source ↗

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