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New small regular graphs of given girth: the cage problem and beyond

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Eleven cage upper bounds fall, with n(4,10) dropping from 384 to 320.

desk verdict Eleven new cage upper bounds, including a big drop for n(4,10), backed by explicit graphs and strong validation; the main residual risk is software verification, not the mathematics. read the letter →

arxiv 2511.07247 v2 pith:VX6K2XPJ submitted 2025-11-10 math.CO cs.DM

Geoffrey Exoo , Jan Goedgebeur , Jorik Jooken , Louis Stubbe , Tibo Van den Eede This is my paper · ORCID
classification math.COcs.DM MSC 05C3505C2505C85
keywords cageproblemextremalgraphtheoryliftsofvoltagegraphsgenerationalgorithmsregulargirthupperboundstabusearch
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's central claim is that four complementary construction methods—exhaustive and heuristic searches over voltage-graph lifts, a hill-climbing base-graph search, and an excision-and-completion technique—produce explicit regular graphs that improve the best known upper bounds for eleven (k,g) pairs of the classical cage problem. If the claimed graphs are correct, n(3,16) ≤ 936, n(3,17) ≤ 2048, n(4,9) ≤ 270, n(4,10) ≤ 320, n(4,11) ≤ 713, n(5,9) ≤ 1116, n(6,11) ≤ 7783, n(8,7) ≤ 774, n(10,7) ≤ 1608, n(12,7) ≤ 2890, and n(14,7) ≤ 4716. Several of these records had stood for 22 years, and the n(4,10) bound is reduced by 64 vertices, from 384 to 320. The same machinery yields new results for related problems: 21 upper bounds for edge-girth-regular graphs, 29 for vertex-girth-regular graphs, 6 for (k,g,g+1)-graphs, and 34 previously undetermined orders in (k,g)-spectra. A reader should care because exact cage numbers are known for only a handful of parameter pairs; each improved bound tightens the gap between the Moore lower bound and the smallest known construction.

What carries the argument

The central construction is the lift of a base graph by a finite group: a small voltage graph whose darts carry group elements, expanded into a covering graph with |Γ| copies of each base vertex. Girth of the lift is characterized by Proposition 1—the shortest closed non-reversing walk whose net voltage is the identity—which lets the search prune voltage assignments without constructing full lifts. Spanning-tree reductions, group and edge automorphisms, and incremental cycle checks cut the assignment space, while exhaustive backtracking and tabu search explore it, and a hill-climbing heuristic grows base graphs and voltages together. The second key mechanism is excision: removing a carefully

What would settle it

Download the eleven posted graphs and run an independent, straightforward routine that checks vertex count, degree sequence, and shortest cycle; any discrepancy—especially any cycle shorter than the claimed girth or a vertex of wrong degree—would refute the corresponding bound. For the six excision-based bounds, re-running the (k,g)-completion on the excised (k,g+1)-cages must reproduce a k-regular graph of girth g on the claimed number of vertices.

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Extended reading notes

Core claim

The paper asserts that by combining voltage-graph lifts with excision, it constructs explicit k-regular graphs of girth g whose orders improve the best known upper bounds for n(3,16), n(3,17), n(4,9), n(4,10), n(4,11), n(5,9), n(6,11), n(8,7), n(10,7), n(12,7), and n(14,7). The most striking single result is n(4,10) ≤ 320, a 64-vertex drop from the long-standing 384 bound. The same computational campaign also settles 34 previously open membership questions in (k,g)-spectra and tightens many bounds for edge-girth-regular, vertex-girth-regular, and (k,g,g+1)-graphs. The graphs themselves are posted online, so each bound is accompanied by a certificate that can be independently checked.

Load-bearing premise

The load-bearing premise is that the software that verifies the order, degree, and girth of each posted graph (and completes the excised graphs) is correct; if any checker has a bug, a claimed upper bound could be false even though the graph file exists.

Editorial extensions

If this is right

  • Eleven upper bounds on n(k,g) are improved; most notably n(4,10) drops from 384 to 320, a 64-vertex improvement.
  • For n(4,11) and n(6,11), the new bounds are the first non-trivial upper bounds; previously only the generic inequality n(k,g) < n(k,g+1) applied.
  • The new (3,16) and (4,10) graphs are bipartite, offering additional support for the open conjecture that every even-girth cage is bipartite.
  • For even degrees 8, 10, 12, 14 and girth 7, the excision technique improves each upper bound by three vertices, showing a systematic saving over previous excision sets.
  • The variant results include 21 new edge-girth-regular, 29 new vertex-girth-regular, 6 new (k,g,g+1)-graph bounds (several tight), and 34 previously undetermined orders in (k,g)-spectra.

Reading between the lines

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

  • The 64-vertex drop for n(4,10) suggests the true value may sit well below 384, possibly within a few dozen vertices of the Moore bound of 242; an exact determination could be attempted by extending these lift searches to orders 280–320.
  • The bipartiteness of the new even-girth graphs suggests that focusing lift searches on bipartite voltage assignments could be a cheap filter for other even-girth cages beyond these cases.
  • The excision improvements for even degrees 8–14 rely on the structural observation that (k,8)-cages have exactly k vertices at distance 2 from both of two distance-4 vertices; if this pattern extends to larger even k, the same excision-plus-completion recipe could improve n(k,8) for all even k.
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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

3 major / 5 minor

Summary. This paper reports new upper bounds for the classical cage problem n(k,g) using computational search methods based on lifts of voltage graphs and on excision. The authors develop four algorithms: an exhaustive backtracking search over voltage assignments (BTA), a tabu search heuristic, a hill climbing heuristic, and an excision procedure that deletes a set of vertices from a known (k,g+1)-graph and completes the remainder to a k-regular graph of girth at least g. Applying these methods, they claim eleven improved upper bounds: n(3,16)≤936, n(3,17)≤2048, n(4,9)≤270, n(4,10)≤320, n(4,11)≤713, n(5,9)≤1116, n(6,11)≤7783, n(8,7)≤774, n(10,7)≤1608, n(12,7)≤2890, and n(14,7)≤4716. They also report improvements for edge-girth-regular, vertex-girth-regular, and (g+1)-cycle-free graphs, as well as 34 newly determined orders in (k,g)-spectra. The paper includes validation of the lift-based generators by disabling pruning optimizations and comparing with three independent implementations.

Significance. If the eleven bounds are correct, this is a significant contribution to the cage problem, several of whose best bounds have been stagnant for years. The improvement n(4,10)≤320, down from the 22-year-old bound 384, is particularly notable. The lift-based framework is methodologically interesting, and the validation strategy—disabling each pruning optimization and matching three independent external generators—is a commendable sanity check. The public repository with code and data is a strength. However, the central claim is purely existential, and the paper's own validation does not cover the excision-based part, so the reader must trust an external completion generator. The paper explicitly acknowledges that the certificates are in the repository rather than in the manuscript.

major comments (3)
  1. [§3.6 and §4.1] Six of the eleven new cage bounds (n(4,11)≤713, n(6,11)≤7783, n(8,7)≤774, n(10,7)≤1608, n(12,7)≤2890, n(14,7)≤4716) rely on the external (k,g)-graph completion generator of [24,38]. The completed graphs are not exhibited in the preprint, and §4.1 validates only the lift-based pipelines (disabling prunes, comparing with three generators); no cross-check of the completion generator is reported. The sentence 'even if an implementation would be incorrect, the certificates ... are available' does not resolve this: if the verifier is faulty, the certificates can be invalid. Please include the completed graphs (e.g., graph6/edge lists) for all six cases and provide/point to an independent verifier (order, degree, girth) with the results stated in the paper.
  2. [§3.6/Fig. 3] The excision sets for the (4,12) and (6,12) cages are described informally: for (4,12) one vertex is removed from 'either N2(u)∩N4(v) or N4(u)∩N2(v)', and for (6,12) three vertices are removed from N3(u,v), without identifying the vertices. Fig. 3 is a schematic drawing, not a unique specification. This makes the constructions irreproducible without the external data. Please specify the exact excised vertex sets, or give the lists of vertices for the particular cages used.
  3. [§3.4.2 and §3.3] The tabu search accepts neighbors with g_nbr = gmin−2 (Table 1), and the paper consistently speaks of 'minimum girth' rather than 'girth exactly g'. The final filter ISINTERESTING is not formally defined, so it is not clear from the paper that every reported graph has girth exactly the target g (a (k,g)-graph) rather than merely at least g (which would not prove the claimed upper bound on n(k,g)). Please state explicitly that each reported graph was verified to have girth exactly g, and describe the verification routine (e.g., the algorithm from [29]).
minor comments (5)
  1. [§2] Typo: 'the the order of r'.
  2. [Algorithms 1 and 4] The function name is written as NOTCANONICAL in Algorithm 1 but ISNOTCANONICAL in Algorithm 4; unify the notation.
  3. [Fig. 4] For Figs. 4a, 4b, 4f the caption says group elements are 'numbered by their index as given by GAP's MultiplicationTable'; this is not self-contained. Include the group presentation or an explicit table of voltages.
  4. [§3.6] 'for even degrees 8≤k≤14' is ambiguous; rephrase as 'for even k with 8≤k≤14'.
  5. [§4] The text says 100 graphs per excision case; please state whether all 100 were independently verified for degree and girth, or only by the completion generator itself.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the eleven upper bounds are existence claims certified by explicit graphs, verified with independent forward checks.

full rationale

The paper's central claims are computational existence results: for each parameter pair it exhibits a graph of stated order, degree and girth, so the inequality n(k,g) <= N follows directly. The derivation chain is not self-referential. Lift-based constructions use Proposition 1 (girth of a lift equals the shortest closed non-reversing walk with net voltage 0) in the forward direction to prune and then verify; the target girth is a search constraint (e.g., Table 1 sets gnbr = gmin - 2 and the final check ISINTERESTING verifies the actual girth on the constructed lift), not an input that forces the output. The excision bounds remove vertices from known (k,g+1)-cages and complete the graph with an external (k,g)-graph generator; the claimed bound is not used as input to the generator, and the resulting graph can be checked directly. Validation against three independent generators and the public graph files provide external checks. The self-citations (e.g., [22] for the lift-girth lemma, [29] for girth/regularity routines, [24,38] for the completion generator) are ordinary references to prior tools and lemmas, not unverified premises that already contain the eleven inequalities. The only evidentiary weak spot is that the preprint itself does not re-run the verifiers on the hosted graph files; that is a verification/correctness risk, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new theoretical entities; its contributions are concrete graphs and algorithms. The only 'free parameters' are search hyperparameters, which do not influence the certificates.

free parameters (2)
  • Search hyperparameters (Table 1) = t_BTA=20s, t_TS=20s, t_TS,init=2s, #φG=200, #φΓ=2000, |L_tabu|=3|Γ|, C=1000, #W=500
    Hand-chosen to balance runtime and search quality; they affect which graphs the heuristics output but do not enter the certificates for the upper bounds.
  • gnbr (neighbor girth in tabu search) = g_min - 2 for cage problem, g_min for regular variants (Table 1)
    An ad hoc search parameter used to prune candidate neighbors; not part of the final graph verification.
assumptions (5)
  • domain assumption A (k,g)-graph exists for all k≥2, g≥3 (Sachs, [45])
    Used to frame the problem; not load-bearing for any specific new upper bound.
  • standard math Proposition 1: girth of a lift equals the length of the shortest closed non-reversing walk with net voltage 0 [22, Lemma 2.1]
    Load-bearing: the girth of all lift-constructed graphs is verified through this characterization.
  • standard math Proposition 2: voltages on a spanning tree can be zeroed with an isomorphic lift [31, p. 91]
    Used to prune the exhaustive voltage-assignment search; not needed for existence of the found graphs.
  • domain assumption Correctness of third-party software: GAP group database, Nauty canonical labeling, multigraph+ generator, and the (k,g)-graph completion generator [24,38]
    The search space and the excision completions are produced by these tools; the paper validates against independent generators but cannot prove them.
  • standard math Inequality n(k,g) < n(k,g+1) [46]
    Used only to state that old bounds for n(4,11) and n(6,11) were trivial.

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Pith. "Pith review of New small regular graphs of given girth: the cage problem and beyond." pith.science (2026). https://pith.science/paper/VX6K2XPJ

@misc{pith2026251107247,
  author       = {Pith},
  title        = {Pith review of: New small regular graphs of given girth: the cage problem and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VX6K2XPJ}},
  note         = {Machine review of arXiv:2511.07247}
}
abstract

The cage problem concerns finding $(k,g)$-graphs, which are $k$-regular graphs with girth $g$, of the smallest possible number of vertices. The central goal is to determine $n(k,g)$, the minimum order of such a graph, and to identify corresponding extremal graphs. In this paper, we study the cage problem and several of its variants from a computational perspective. Four complementary graph generation algorithms are developed based on exhaustive generation of lifts, a tabu search heuristic, a hill climbing heuristic and excision techniques. Using these methods, we establish new upper bounds for eleven cases of the classical cage problem: $n(3,16) \leq 936$, $n(3,17) \leq 2048$, $n(4,9) \leq 270$, $n(4,10) \leq 320$, $n(4,11) \leq 713$, $n(5,9) \leq 1116$, $n(6,11) \leq 7783$, $n(8,7) \leq 774$, $n(10,7) \leq 1608$, $n(12,7) \leq 2890$ and $n(14,7) \leq 4716$. Notably, our results improve upon several of the best-known bounds, some of which have stood unchanged for 22 years. Moreover, the improvement for $n(4,10)$, from the longstanding upper bound of 384 down to 320, is surprising and constitutes a substantial improvement. While the main focus is on the cage problem, we also adapted our algorithms for variants of the cage problem that received attention in the literature. For these variants, additional improvements are obtained, further narrowing the gaps between known lower and upper bounds.

Figures

Figures reproduced from arXiv: 2511.07247 by the authors.

Figure 1
Figure 1. A lift of the dumbbell graph with the group [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The resulting 3-regular graphs with parallel edges, loops and semi-edges obtained through multigraph+. Only degree distributions yielding graphs are shown. The graphs marked in red are isomorphic. Semi-edge constraints For the lift to be interpretable as an undirected graph, the voltage for each dart corresponding with a semi-edge must be its own inverse. Moreover, these darts should not be assigned 0Γ as this would… view at source ↗
Figure 3
Figure 3. The excised set of vertices indicated in red from the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The base graphs and groups that resulted in a new upper bound for the cage problem. Darts corresponding to [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

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