{"id":"13c33d44-f827-4320-b843-91e33178afff","arxiv_id":"1908.09361","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every integer m >= 1, there is a connected tetravalent half-arc-transitive graph with vertex stabilizer D8^2 x C2^m, built as a coset graph on the alternating group A_{2m+6}.","lead":"This paper constructs an infinite family of tetravalent half-arc-transitive graphs, answering an open question about possible vertex stabilizers of order a power of two. The graphs are explicit coset graphs on alternating groups and come with control over their automorphism groups and associated digraphs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's automorphism-group proof rests on unproved fixed-point counts in Lemmas 3.8 and 3.9, delegated to the paper itself; an independent check of these counts is needed.","rationale":"The reader's conditional verdict is justified. The construction is explicit and the high-level group-theoretic framework is standard, which gives the theorem plausibility, but the proof is not self-contained at its most delicate numerical point: Section 3 states Lemmas 3.8 and 3.9 without proof and cites [55], the paper itself, for 'tedious calculation'. These lemmas are not decorative: Proposition 5.3 uses them to rule out every nontrivial automorphism of the connection set, and Proposition 6.1 obtains Aut(Γ_m) = Alt(H) through [18, Theorem 1.1] only after that elimination. The connectivity proof likewise imports Lemmas 4.1–4.4 from the same self-reference. The concern is not that the counts are false; it is that the core numerical content of the theorem is not verifiable from the reviewed text. An independent small-m computation would settle the matter. Because the reader already assigned CONDITIONAL, this stress-test does not move the verdict; it reinforces the condition.","tokens_in":21220,"tokens_out":7968,"duration_ms":75030,"concrete_test":"Implement Construction 3.3 literally in GAP or Magma for m = 1, 2, 3, 4, 5: construct H, the automorphism x by equation (2), the permutation y by equation (3), and z = R(f)yR(f c d e_m^m); then brute-force compute |Fix(xyxz)| and |Fix((xyxz)^2)| (Lemma 3.8) and the seven intersection sizes in Lemma 3.9(a)–(g), covering all parity classes modulo 4. Verify that the values equal 3, 2^{m+3}, 2^{m+1}, and the listed (1,2)/(2,1) patterns. If every computed value matches, the omitted calculations are supported; any mismatch would directly refute Proposition 5.3 as written and hence the claimed conclusion of Theorem 1.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem depends on proving Aut(Alt(H)_1, S) = 1 in Proposition 5.3 and then Aut(Γ_m) = Alt(H) in Proposition 6.1. In Proposition 5.3, every possible image of xy under a hypothetical nontrivial automorphism σ is eliminated only through the numerical fixed-point assertions in Lemmas 3.8 and 3.9: Lemma 3.8 gives |Fix(xyxz)| = 3 for even m and |Fix((xyxz)^2)| = 3 for odd m, while Lemma 3.9 supplies intersection sizes 2^{m+3}, 2^{m+1}, and the four parity patterns (1,2), (1,2), (2,1), (2,1) in parts (d)–(g). A single miscount could admit a nontrivial σ, break Aut(Alt(H)_1, S) = 1, and invalidate the application of [18, Theorem 1.1] that forces Aut(Γ_m) = Alt(H) in Proposition 6.1. The text states that these lemmas are given 'without proof, as it is tedious calculation and can be found in [55]', and reference [55] is this same arXiv paper; the reviewed version does not contain the promised appendix. The same self-referential delegation applies to Lemmas 3.7, 3.10, and 4.1–4.4 used for connectivity and radius, but Lemmas 3.8 and 3.9 are the most load-bearing because they are the only numerical inputs in the automorphism-group elimination.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every positive integer m, a connected tetravalent half-arc-transitive graph Γ_m with vertex stabilizer D_8^2 × C_2^m. The construction is a coset graph Cos(Alt(H), R(H), R(H){xy,(xy)^{-1}}R(H)), where H is a group of order 2^{m+6}. The author proves that Γ_m is a nonnormal Cayley graph on the alternating group A_{2m+6−1}, that Aut(Γ_m) ≅ A_{2m+6}, that Γ_m is loosely attached of radius 6, and that its two associated digraphs are (m+6)-arc-transitive and not self-reverse. This is claimed as an affirmative answer to Question 1.1 for all s ≥ 3 except s = 5, which is handled via a separate cover construction. The main proof is group-theoretic and not computer-assisted, but several technical lemmas are stated without proof and deferred to the author's own arXiv paper, which is the same paper under review.","tokens_in":21616,"tokens_out":4398,"duration_ms":41643,"significance":"If the construction and proof are correct, this is a substantial contribution. It resolves a long-standing open question, provides the first infinite family of connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers of arbitrarily large 2-power order, and yields new examples relevant to nonnormal Cayley graphs on simple groups, quasiprimitive normal quotient analysis, and highly arc-transitive non-self-reverse digraphs. The explicit, parameterized construction is a strength, and several parts of the paper (e.g., the connectivity argument via fixed-point ratios and the normal quotient analysis in Proposition 6.1) are carefully reasoned. However, the paper currently lacks proofs of the key fixed-point lemmas, which are load-bearing for the automorphism group determination. Until those calculations are supplied in a verifiable form, the main theorem cannot be considered established.","major_comments":[{"comment":"Lemmas 3.8 and 3.9 are stated without proof, with the text saying 'it is tedious calculation and can be found in [55]'. Reference [55] is the arXiv identifier of this same paper. In the version under review no appendix or supplementary material containing these calculations is present. These fixed-point counts are load-bearing: Lemma 3.8 is used in Proposition 4.5 to rule out the affine case, and together with Lemma 3.9 it supplies the numerical contradictions in every case of Proposition 5.3, which establishes Aut(Alt(H)_1,S)=1. Proposition 6.1 then depends on this to force Aut(Γ_m)=Alt(H). Without an independent proof or a formally certified computation, Theorem 1.2(c) and (d) are not established.","section":"Section 3, Lemmas 3.8 and 3.9"},{"comment":"The same self-referential deferral applies to Lemmas 3.4–3.7, 3.10, and 4.1–4.4, which are also said to be proved in [55]. These are not merely cosmetic: Lemma 3.7 supplies the fixed-point sets and the order |yz|=6 used in Proposition 4.5 and Proposition 5.3, Lemma 3.10 is used in Proposition 6.2 to show attachment number 1, and Lemmas 4.1–4.4 are essential in the connectivity proof. The paper should provide complete proofs of all these lemmas, or at minimum place the promised Magma verification code and its certified output in an appendix that is actually included in the manuscript.","section":"Sections 3 and 4, Lemmas 3.4–3.7, 3.10, 4.1–4.4"}],"minor_comments":[{"comment":"The sentence 'The lemmas are stated below without proof, as it is tedious calculation and can be found in [55]' should be revised for grammar and, more importantly, should not point to the paper itself as the location of the proofs.","section":"Section 3, p. 9"},{"comment":"The text before Proposition 6.3 says that the proof 'is given by an anonymous referee'. This attribution is unusual and should be removed; the proof should be presented as an integral part of the paper without external authorship statements.","section":"Section 6, Proposition 6.3"},{"comment":"The notation table is helpful but would be improved by explicitly stating that e_i is, by convention, the identity for i ≤ 0 before it is used in the action formulas; currently this convention appears only in the running text after the table.","section":"Section 3, Table of notation"},{"comment":"The phrase 'gives the affirmative answer to Question 1.1 for s ≠ 5' is slightly imprecise because the construction covers s = m + 6, so for s = 6,7,... it gives infinitely many values; please clarify that the remaining case s = 5 is covered by the cited result of [52].","section":"Introduction, p. 3"}],"recommendation":"major_revision","confidential_remarks":"The self-referential nature of reference [55] is a serious flaw in the current version: the paper delegates its most technical assertions to itself, so the manuscript is not verifiable as submitted. If the author can provide full proofs or a reproducible computational certificate for the deferred lemmas, the construction and the overall structure of the proof are plausible and would justify publication in a strong journal. I urge the editor to require that the appendix be included in any revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real: for every m ≥ 1 the paper constructs a connected tetravalent half-arc-transitive graph Γ_m with vertex stabilizer D8^2 × C2^m, giving the first infinite family with nonabelian vertex stabilizer and answering Question 1.1. On top of that, Γ_m is a nonnormal Cayley graph on A_{2m+6−1}, is loosely attached with radius 6, and its two oriented digraphs are (m+6)-arc-transitive and not self-reverse. These extra properties are not just decoration; they connect to several open problems in the area. The construction itself is explicit and the overall proof strategy is sound, using standard tools from permutation groups and coset graphs. The paper is also honest about the fact that reference [52] independently answered Question 1.1 via a different construction with stabilizer D8 × C2^m; the present family is still new because of the stabilizer shape and the bundle of properties.\n\nThe soft spot is exactly where the stress-test note points. Several lemmas are stated without proof, including the load-bearing fixed-point counts in Lemmas 3.8 and 3.9. These counts are the numerical inputs that eliminate automorphisms in Proposition 5.3 and force Aut(Γ_m) = Alt(H) in Proposition 6.1. The text says the proofs are tedious and can be found in [55], and [55] is this same arXiv paper. In the version I have, the promised appendix is not present. So the proof as written is incomplete: a referee cannot verify the central argument without redoing those calculations independently. This is not a minor omission; it is load-bearing. The same self-delegation appears for Lemmas 3.4–3.7, 3.10, and 4.1–4.4, but the two fixed-point lemmas are the ones that matter most.\n\nIf the omitted counts are correct, the theorem likely stands. The rest of the proof is careful and the architecture is credible. The issue is purely that the manuscript does not currently contain the verification. This is fixable by supplying the appendix or full proofs.\n\nWho should read this: algebraic graph theorists working on half-arc-transitive graphs or Cayley graph automorphisms. It deserves peer review, but a serious referee should insist on seeing the missing calculations before acceptance. I would not cite the current version as a complete proof; I would wait for the published version with the appendix. The paper is worth engaging with, and the underlying idea is good.","headline":"A genuinely new infinite family of tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, but the current text leaves key fixed-point calculations to its own missing appendix.","tokens_in":22098,"tokens_out":2062,"would_cite":true,"duration_ms":20136,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B25","05C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every integer $m\\geq1$, there is a connected tetravalent half-arc-transitive graph whose automorphism group is the alternating group $A_{2m+6}$ and whose vertex stabilizer is the nonabelian group $D_8^2\\times C_2^m$; hence these…","keywords":["half-arc-transitive graphs","tetravalent graphs","vertex stabilizers","nonabelian groups","Cayley graphs","nonnormal Cayley graphs","alternating groups","arc-transitive digraphs"],"falsifier":"Compute the asserted fixed-point values for a small concrete case, say $m=1$ or $m=2$, directly from the definitions of $x,y,z$; if $|\\mathrm{Fix}(xyxz)|\\neq3$ for even $m$, or $|\\mathrm{Fix}(yz)\\cap\\mathrm{Fix}(yz)^{xyzx^{-1}}|\\neq2^{m+3}$, or any parity-dependent value in Lemma 3.9(d)--(g) differs, then the proof of Propositions 5.3 and 6.1 collapses and the theorem fails for that $m$.","tokens_in":21035,"feed_emoji":"","tokens_out":14371,"duration_ms":123050,"temperature":0.7,"pith_summary":"This paper proves that tetravalent half-arc-transitive graphs can have nonabelian vertex stabilizers of arbitrarily large $2$-power order. For every positive integer $m$ it constructs a connected tetravalent half-arc-transitive graph $\\Gamma_m$ whose full automorphism group is the alternating group $A_{2m+6}$ and whose vertex stabilizer is $D_8^2\\times C_2^m$, a nonabelian group of order $2^{m+6}$. This answers a question open since 2015, when only six connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers were known. The same graphs are nonnormal Cayley graphs on the alternating group $A_{2m+5}$ with trivial connection-set stabilizer, and their two orientation digraphs are $(m+6)$-arc-transitive and not self-reverse.","feed_headline":"Half-arc-transitive graphs now exist with unbounded vertex stabilizers","feed_subtitle":"An infinite family answers the long-open question of whether these stabilizers can have arbitrarily large 2-power order.","key_machinery":"The central object is the coset graph $\\Gamma_m=\\mathrm{Cos}(\\mathrm{Alt}(H),R(H),R(H)\\{xy,(xy)^{-1}\\}R(H))$, where $H=D_8^2\\times C_2^m$, $R(H)$ is the regular right-multiplication action of $H$, and $x,y,z$ are specific permutations of $H$ fixing the identity: $x$ is an automorphism of order $4$, $y$ is an involution, and $z=R(f)yR(f c d e_m^m)$ is a derived involution. The proof of connectivity shows $\\langle R(H),xy\\rangle=\\mathrm{Alt}(H)$ by ruling out affine and product-action primitive groups, and the identity $|yz|=6$ forces the alternating cycles to have length $6$ while the intersection of the two alternating cycles through a vertex is just that vertex, giving attachment number $1$. The fixed-point counts of $yz$, $xyxz$, and their conjugates, recorded in Lemmas 3.7--3.10, are the load-bearing calculations that eliminate every potential automorphism of the Cayley graph and force $\\mathrm{Aut}(\\Gamma_m)=\\mathrm{Alt}(H)$.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.2: for every $m\\geq1$ there is a graph $\\Gamma_m$ that is a connected nonnormal Cayley graph on the alternating group $A_{2m+5}$ with $\\mathrm{Aut}(A_{2m+5},S)=1$, a loosely attached tetravalent half-arc-transitive graph of radius $6$, with $\\mathrm{Aut}(\\Gamma_m)\\cong A_{2m+6}$, vertex stabilizer $D_8^2\\times C_2^m$, and two orientation digraphs that are both $(m+6)$-arc-transitive and not self-reverse. The proof is not computer-assisted in the paper, though the paper notes that computations can verify the cases for small $m$.","pith_inferences":["The same double-coset construction may extend to other nonabelian $2$-groups $H$ that admit an automorphism of order $4$ and an involution $y$ with cycle structure matching the lemmas; testing the next-smallest candidates would show how rigid the $D_8^2\\times C_2^m$ form is.","If the fixed-point counts in Lemmas 3.8 and 3.9 can be expressed as closed formulas in $m$, the proof would become self-contained and might explain why the vertex stabilizer is exactly a direct product of two copies of $D_8$ with an elementary abelian $2$-group.","The graphs show that the condition $\\mathrm{Aut}(G,S)=1$ is far from sufficient for a Cayley graph on a simple group to be a graphical regular representation; studying the action of $\\mathrm{Aut}(\\Gamma_m)$ on the cosets of $R(G)$ may clarify exactly when normality fails."],"forward_implications":["Together with the earlier examples with stabilizers $D_8$, $D_8\\times C_2$ and $D_8^2$, the family answers Question 1.1: for every integer $s\\geq3$ there is a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$.","Each $\\Gamma_m$ is a nonnormal Cayley graph on the simple group $A_{2m+5}$ with $\\mathrm{Aut}(A_{2m+5},S)=1$, giving an infinite family of connected tetravalent edge-transitive nonnormal Cayley graphs on nonabelian simple groups.","Since $\\mathrm{Aut}(\\Gamma_m)\\cong A_{2m+6}$ is simple, each pair $(\\Gamma_m,\\mathrm{Aut}(\\Gamma_m))$ is a quasiprimitive example in the normal-quotient analysis of tetravalent half-arc-transitive graphs.","The two orientation digraphs $D_1(\\Gamma_m)$ and $D_2(\\Gamma_m)$ are both $(m+6)$-arc-transitive and not self-reverse, so non-self-reverse digraphs occur with arbitrarily large arc-transitivity."],"supporting_citations":[{"why":"posed the existence question for nonabelian vertex stabilizers of order $2^s$ and supplied the examples that the new family extends.","marker":"[13]"},{"why":"constructed the immediate predecessors with stabilizers $D_8^2$ and a nonabelian group of order $128$, providing the pattern this construction generalizes.","marker":"[49]"},{"why":"built the infinite family of cubic nonnormal Cayley graphs on alternating groups whose approach inspires the present Cayley-graph construction.","marker":"[7]"},{"why":"gives the classification theorem on automorphism groups of nonnormal Cayley graphs of simple groups used to pin down $\\mathrm{Aut}(\\Gamma_m)$.","marker":"[18]"},{"why":"supplies the minimal-degree bound for primitive permutation groups used to rule out affine and product-action cases in the connectivity proof.","marker":"[27]"},{"why":"classifies subgroups of prime power index in simple groups, used to identify the socle of $\\mathrm{Aut}(\\Gamma_m)$ as $A_{2m+6}$.","marker":"[26]"},{"why":"established the alternating-cycle and attachment-number theory for tetravalent half-arc-transitive graphs that underlies the radius-6 and loosely attached conclusions.","marker":"[34]"},{"why":"contains the tedious fixed-point calculations behind Lemmas 3.4--3.10, on which the elimination of unwanted automorphisms depends.","marker":"[55]"}],"fun_headline_variants":["Infinite family of half-arc-transitive graphs with unbounded stabilizers","Arbitrarily large nonabelian stabilizers in half-arc-transitive graphs","2-power stabilizer question answered with infinite family","Tetravalent half-arc-transitive graphs: unbounded nonabelian stabilizers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the fixed-point counts asserted in Lemmas 3.8 and 3.9 are correct, for example $|\\mathrm{Fix}(yz)\\cap\\mathrm{Fix}(yz)^{xyzx^{-1}}|=2^{m+3}$ and the parity-dependent values in parts (d)--(g); the paper states these lemmas without proof and refers to its own extended version for the calculations, and if any one count is wrong the arguments excluding unwanted automorphisms in Propositions 5.3 and 6.1 break down.","fun_headline_variants_meta":{"raw":{"variants":["Infinite family of half-arc-transitive graphs with unbounded stabilizers","Arbitrarily large nonabelian stabilizers in half-arc-transitive graphs","2-power stabilizer question answered with infinite family","Tetravalent half-arc-transitive graphs: unbounded nonabelian stabilizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001254,"raw_usage":{"total_tokens":5132,"prompt_tokens":929,"completion_tokens":4203,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":4121}},"tokens_in":545,"tokens_out":4203,"duration_ms":32358,"temperature":1.0,"reasoning_tokens":4121,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:13:39.583961+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the asserted fixed-point values for a small concrete case, say $m=1$ or $m=2$, directly from the definitions of $x,y,z$; if $|\\mathrm{Fix}(xyxz)|\\neq3$ for even $m$, or $|\\mathrm{Fix}(yz)\\cap\\mathrm{Fix}(yz)^{xyzx^{-1}}|\\neq2^{m+3}$, or any parity-dependent value in Lemma 3.9(d)--(g) differs, then the proof of Propositions 5.3 and 6.1 collapses and the theorem fails for that $m$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"posed the existence question for nonabelian vertex stabilizers of order $2^s$ and supplied the examples that the new family extends."},{"cited_title":"Spiga, Constructing half-arc-transitive graphs of valency four with prescribed vertex stabilizers, Graphs Combin","cited_arxiv_id":null,"evidence_quote":"constructed the immediate predecessors with stabilizers $D_8^2$ and a nonabelian group of order $128$, providing the pattern this construction generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"built the infinite family of cubic nonnormal Cayley graphs on alternating groups whose approach inspires the present Cayley-graph construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the classification theorem on automorphism groups of nonnormal Cayley graphs of simple groups used to pin down $\\mathrm{Aut}(\\Gamma_m)$."},{"cited_title":"Guralnick and K","cited_arxiv_id":null,"evidence_quote":"supplies the minimal-degree bound for primitive permutation groups used to rule out affine and product-action cases in the connectivity proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"classifies subgroups of prime power index in simple groups, used to identify the socle of $\\mathrm{Aut}(\\Gamma_m)$ as $A_{2m+6}$."},{"cited_title":"Maruˇ siˇ c, Half-transitive group actions on finite graphs of valency 4, J","cited_arxiv_id":null,"evidence_quote":"established the alternating-cycle and attachment-number theory for tetravalent half-arc-transitive graphs that underlies the radius-6 and loosely attached conclusions."},{"cited_title":"Tetravalent half-arc-transitive graphs with unbounded nonabelian vertex stabilizers","cited_arxiv_id":"1908.09361","evidence_quote":"contains the tedious fixed-point calculations behind Lemmas 3.4--3.10, on which the elimination of unwanted automorphisms depends."}],"review_version":1}