{"id":"620257b1-8afb-42ca-8552-03622abdc2d1","arxiv_id":"1908.03115","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every bipartite graph G and every s >= 2, the Castelnuovo-Mumford regularity of the s-th power of its edge ideal is at most 2s + reg(I(G)) - 2, and this bound is tight.","lead":"Edge ideals of bipartite graphs have powers whose Castelnuovo-Mumford regularity is bounded by 2s plus the regularity of the original ideal minus 2. The proof combines a topological bound for the square with an algebraic induction, and the bound is shown to be sharp for every s.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the Castelnuovo-Mumford regularity of powers of edge ideals. The main results are: (i) for any finite simple graph G, reg(I(G)^2) ≤ reg(I(G)) + 2; and (ii) for any bipartite graph G, reg(I(G)^s) ≤ 2s + reg(I(G)) − 2 for all s ≥ 2. The proof combines Hochster's formula with a topological inequality on clique complexes (Theorem 3.1) and an algebraic induction using colon ideals, polarization, and a theorem of Alilooee and Banerjee (Theorem 2.5). The bipartite bound is best possible, as complete bipartite graphs attain equality.","tokens_in":8351,"tokens_out":35018,"duration_ms":342450,"significance":"The bipartite statement (Theorem 1.1(ii)) is a substantial result: it proves a conjecture of Banerjee and others that the regularity of powers of bipartite edge ideals is bounded by 2s + reg(I(G)) − 2, and it shows the bound is tight. The proof introduces a new topological technique, suspension of clique complexes, to handle the base case s = 2. The paper is clearly organized and the algebraic induction for part (ii) is well structured. However, two gaps need to be addressed: a missing justification of the suspension identification in Theorem 3.1, and an erroneous algebraic inequality in the proof of part (i) for non-bipartite graphs.","major_comments":[{"comment":"The identification H_l(Σ_{a,b}(Δ'[W_C])) = H_l(Δ[{a,b} ∪ W_C]) is asserted without proof. This equality is load-bearing: it is used to conclude that the third term in the first Mayer-Vietoris sequence vanishes, which is essential for the base case reg(I(G)^2 : ab) ≤ reg(I(G)). The equality is not immediate from the definitions; the authors should provide the argument that because W_C ⊆ C = A∩B, every vertex of W_C is non-adjacent in G to both a and b, so the added edges in G' do not affect W_C, and the induced complex Δ[{a,b} ∪ W_C] is exactly the suspension of Δ'[W_C]. The same reasoning is needed again in Claim Two.","section":"Section 3, Theorem 3.1, Claim One"},{"comment":"The step 'Let J' := I(G[V \\ N(u)]), then (J : u) = J' + (variables) so reg(J : u) ≤ reg(J')' is false as stated. For example, take G to be the triangle with a leaf: vertices a,b,c,d and edges ab, ac, bc, cd. Then u = c lies in N(a)∩N(b), J = I(G), J' = I(G[{c}]) = 0, but (J : c) = (a,b,d) has regularity 1, so reg(J : u) ≤ reg(J') fails. This invalidates the proof of Theorem 1.1(i) for graphs where a common neighbor of an edge is adjacent to all other vertices. Since the bipartite case has N(a)∩N(b) = ∅, this error does not directly affect Theorem 1.1(ii), but the statement of (i) requires a corrected argument or a suitable restriction.","section":"Section 3, proof of Theorem 1.1(i)"}],"minor_comments":[{"comment":"The notation 'A ∪_C B' is not defined; it should be A ∪ B (or explained).","section":"Section 3, Theorem 3.1"},{"comment":"The regularity of a simplicial complex reg(Δ) is used without definition; it should be defined as max{l+2 : H_l(Δ[W]) ≠ 0}, consistent with Theorem 2.7.","section":"Section 3, Theorem 3.1"},{"comment":"In the short exact sequences, the index n is used where k should be, and the last sequence should involve u_k, not u_n.","section":"Section 3, proof of Theorem 1.1(i)"},{"comment":"There are several typos, including 'the varibles' in Section 2 and 'Froberg' in the introduction (should be 'Fröberg').","section":"Throughout"},{"comment":"Reference [26] is incomplete; it is listed as 'R. Woodrofe, J. Commut. Algebra 6, no. 2 (2014), 287–304' without a title.","section":"References"},{"comment":"In the discussion before Conjecture 4.3, the expression 'reg((I(G)^2 : ab) = 3' is missing a closing parenthesis.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The main bipartite theorem is likely correct and the gaps appear fixable. The topological identification in Theorem 3.1 can be justified with a short argument. The algebraic error in part (i) requires a more substantial repair; if the authors cannot fix the general case, they might restrict part (i) to bipartite graphs, which is all that is needed for the main theorem. I recommend asking for a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper proves the conjectured bound reg(I(G)^s) ≤ 2s + reg I(G) − 2 for all bipartite graphs, and proves the s=2 case for all graphs. The bound is sharp, and it improves on the earlier bound by Jayanthan–Narayanan–Selvaraja that involved the cochordal number. That is a genuine advance, not a routine extension.\n\nWhat is new is the suspension argument in Theorem 3.1, which gives reg(∆') ≤ reg(∆) where ∆' comes from the graph operation adding edges between neighborhoods of an edge. Using that, the algebraic part for s>2 is a clean induction via colon ideals. The proof for s=2 is topological (Hochster's formula plus Mayer–Vietoris), which is the part that needs careful reading.\n\nThe soft spot is exactly where the reader flagged: Claim One in Theorem 3.1. The equality H_l(Σ_{a,b}(∆'[WC])) = H_l(∆[{a,b}∪WC]) is stated in one line. It is true: on WC the complexes ∆ and ∆' agree, and the suspension join with the two vertices a,b gives the induced subcomplex. But the paper does not spell out why, and a non-specialist could get stuck. The same for the subsequent vanishing H_{l−1}(∆'[WC]) = 0. This is an exposition problem, not a mathematical gap, as far as I can tell. I would ask the authors to expand this claim in the next version.\n\nThere is no circularity concern. The cited theorems by the first author are published and independent, and the paper introduces no fitted constants. The example in the introduction showing sharpness is fine.\n\nOne minor thing: the text has a few typos in the proof (e.g. Wc vs WC), and the notation in the decomposition of W is a bit dense. Nothing that undermines the argument.\n\nWho should read this: anyone working on regularity of powers of edge ideals. It settles a natural conjecture for bipartite graphs and gives the first universal bound for s=2. It deserves a serious referee; I'd send it out. The referee should mainly verify Theorem 3.1 line by line, and the rest will go through.\n\nMy recommendation: accept with minor revision, or at least send to peer review. If I were the editor, I would not desk reject.","headline":"Sharp bound for bipartite edge ideals and a new s=2 case; the topological proof is compressed but the mathematics looks right.","tokens_in":8854,"tokens_out":3937,"would_cite":true,"duration_ms":37960,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","05E40","13F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves sharp linear bounds on the regularity of powers of bipartite edge ideals, with a square bound valid for all graphs.","keywords":["Castelnuovo-Mumford regularity","edge ideals","powers of ideals","bipartite graphs","simplicial suspension","Mayer-Vietoris sequence","colon ideals","combinatorial commutative algebra"],"falsifier":"Take any graph $G$, an edge $ab$, and the induced subcomplexes $A$, $B$, $C$ as in Theorem 3.1; choose $W$ with $\\widetilde H_\\ell(\\Delta'[W])\\ne 0$ and set $W_C=A\\cap B\\cap W$. Compute $\\widetilde H_\\ell(\\Delta[\\{a,b\\}\\cup W_C])$ and compare it with $\\widetilde H_\\ell(\\Sigma_{a,b}(\\Delta'[W_C]))$. A mismatch for any one such triple would invalidate Claim One and the proof of the base case $\\operatorname{reg}(I(G)^2:ab)\\le \\operatorname{reg}(I(G))$.","tokens_in":8240,"feed_emoji":"📐","tokens_out":8937,"duration_ms":88570,"temperature":0.7,"pith_summary":"This paper studies how the Castelnuovo-Mumford regularity of an edge ideal grows when the ideal is raised to powers. It proves that for every finite simple graph $G$, $\\operatorname{reg}(I(G)^2) \\le \\operatorname{reg}(I(G))+2$, and that when $G$ is bipartite, $\\operatorname{reg}(I(G)^s) \\le 2s+\\operatorname{reg}(I(G))-2$ for all $s\\ge2$. The bound is best possible: complete bipartite graphs attain equality. The square case is proved by a topological argument built on suspension of simplicial complexes, and the higher-power case follows by passing through colon ideals and short exact sequences. The result confirms the conjectured upper bound for every bipartite graph and supports the broader conjecture that the same inequality holds for all graphs.","feed_headline":"Edge-ideal squares add at most 2 to regularity","feed_subtitle":"For bipartite graphs, higher powers obey the sharp bound reg(I^s) ≤ 2s + reg(I) − 2.","key_machinery":"The load-bearing object is the suspension of a simplicial complex, $\\Sigma_{a,b}\\Delta = \\Delta * \\{\\{a\\},\\{b\\},\\varnothing\\}$, whose geometric realization is the topological suspension. Theorem 3.1 compares $\\Delta=\\operatorname{cl}(G^c)$ with the clique complex $\\Delta'$ of the graph $G'$ obtained from $G$ by connecting every neighbor of $a$ to every neighbor of $b$, and proves $\\operatorname{reg}(\\Delta')\\le \\operatorname{reg}(\\Delta)$. This topological inequality controls the colon ideal $(I(G)^2:ab)$, and an algebraic induction based on short exact sequences and Theorem 2.3 carries the bound from squares to all powers.","core_discovery":"The central claim is Theorem 1.1. For any finite simple graph $G$ with edge ideal $I(G)$, squaring raises regularity by at most $2$: $\\operatorname{reg}(I(G)^2) \\le \\operatorname{reg}(I(G))+2$. If $G$ is bipartite, then for every $s\\ge2$, $\\operatorname{reg}(I(G)^s) \\le 2s+\\operatorname{reg}(I(G))-2$. This second inequality is the conjectured universal bound specialized to bipartite graphs, and it is sharp because complete bipartite graphs satisfy $\\operatorname{reg}(I(G)^s)=2s$. The proof of the square case is topological: an auxiliary clique complex $\\Delta'$ built from $\\Delta=\\operatorname{cl}(G^c)$ by joining the neighbors of an edge's endpoints is shown to have regularity no larger than $\\Delta$ (Theorem 3.1). The passage from squares to all powers uses colon ideals, with the bipartite structure guaranteeing that the relevant colons remain edge ideals on the same bipartition.","pith_inferences":["Editorial: The square bound $\\operatorname{reg}(I(G)^2)\\le \\operatorname{reg}(I(G))+2$ is proved without the bipartite assumption, so if the gap noted below is repaired, the topological suspension strategy may extend the bound to higher powers for classes of graphs beyond bipartite ones.","Editorial: The iteration from squares to all powers depends on the bipartite colon property; finding an analogue of that property for other graph classes would immediately yield the same sharp bound there.","Editorial: The paper's closing example of a flag-no-square dunce-hat triangulation gives a concrete computational experiment: evaluating $\\operatorname{reg}(I(G)^2)$ for that graph would test the related conjecture discussed in Section 4, a computation the authors leave open."],"forward_implications":["For bipartite graphs, the regularity sequence of powers satisfies $\\operatorname{reg}(I(G)^s) \\le 2s+\\operatorname{reg}(I(G))-2$, matching the conjectured universal bound.","Complete bipartite graphs attain equality, so the bound cannot be improved to a smaller intercept for bipartite graphs purely in terms of $\\operatorname{reg}(I(G))$.","The theorem verifies the longstanding inequality $b(I(G))\\le \\operatorname{reg}(I(G))-2$ for all bipartite graphs, where $b(I(G))$ is the eventual intercept of the linear regularity sequence.","For every finite simple graph, not only bipartite ones, squaring the edge ideal raises regularity by at most $2$, giving the base step needed to test the general conjecture for higher powers."],"supporting_citations":[{"why":"Supplies the short exact sequence, Theorem 2.2, and Theorem 2.3 used to reduce powers of edge ideals to colon ideals.","marker":"[2]"},{"why":"Gives the bipartite colon property: colons at edge generators remain edge ideals on the same bipartition, enabling iteration over $s$.","marker":"[1]"},{"why":"Hochster's formula re-expresses regularity as the maximum over reduced homology groups of induced subcomplexes, anchoring the topological proof.","marker":"[20]"},{"why":"Provides the fact that regularity of a sum of ideals in disjoint variables is the sum of regularities minus one, used in the colon computation for $(L:u)$.","marker":"[26]"},{"why":"The earlier bipartite bound stated in the introduction, which Theorem 1.1 improves.","marker":"[16]"},{"why":"Fr\\\"oberg's characterization of edge ideals of regularity $2$, used in the sharpness discussion.","marker":"[12]"},{"why":"Shows powers of cochordal edge ideals have linear resolutions, giving $\\operatorname{reg}(I(G)^s)=2s$ for complete bipartite graphs and hence equality in the bound.","marker":"[15]"}],"fun_headline_variants":["Bipartite edge ideals: sharp regularity bound for all powers","Squaring edge ideals: regularity rises at most 2","Sharp bound for powers of bipartite edge ideals","Suspension yields sharp edge-ideal power bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, in Claim One of Theorem 3.1, the subcomplex on the vertex set $\\{a,b\\}\\cup W_C$ is exactly the suspension $\\Sigma_{a,b}(\\Delta'[W_C])$; this equality is asserted without proof, and the vanishing of the homology that carries the base case depends on it.","fun_headline_variants_meta":{"raw":{"variants":["Bipartite edge ideals: sharp regularity bound for all powers","Squaring edge ideals: regularity rises at most 2","Sharp bound for powers of bipartite edge ideals","Suspension yields sharp edge-ideal power bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000925,"raw_usage":{"total_tokens":3901,"prompt_tokens":822,"completion_tokens":3079,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":3013}},"tokens_in":438,"tokens_out":3079,"duration_ms":27027,"temperature":1.0,"reasoning_tokens":3013,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:24:32.561515+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any graph $G$, an edge $ab$, and the induced subcomplexes $A$, $B$, $C$ as in Theorem 3.1; choose $W$ with $\\widetilde H_\\ell(\\Delta'[W])\\ne 0$ and set $W_C=A\\cap B\\cap W$. Compute $\\widetilde H_\\ell(\\Delta[\\{a,b\\}\\cup W_C])$ and compare it with $\\widetilde H_\\ell(\\Sigma_{a,b}(\\Delta'[W_C]))$. A mismatch for any one such triple would invalidate Claim One and the proof of the base case $\\operatorname{reg}(I(G)^2:ab)\\le \\operatorname{reg}(I(G))$.","supporting_citations":[{"cited_title":"Banerjee, Regularity of Powers of Edge Ideals, J","cited_arxiv_id":null,"evidence_quote":"Supplies the short exact sequence, Theorem 2.2, and Theorem 2.3 used to reduce powers of edge ideals to colon ideals."},{"cited_title":"Alilooee, A","cited_arxiv_id":null,"evidence_quote":"Gives the bipartite colon property: colons at edge generators remain edge ideals on the same bipartition, enabling iteration over $s$."},{"cited_title":"Miller and B","cited_arxiv_id":null,"evidence_quote":"Hochster's formula re-expresses regularity as the maximum over reduced homology groups of induced subcomplexes, anchoring the topological proof."},{"cited_title":"Woodrofe, J","cited_arxiv_id":null,"evidence_quote":"Provides the fact that regularity of a sum of ideals in disjoint variables is the sum of regularities minus one, used in the colon computation for $(L:u)$."},{"cited_title":"Jayanthan, N","cited_arxiv_id":null,"evidence_quote":"The earlier bipartite bound stated in the introduction, which Theorem 1.1 improves."},{"cited_title":"Fr¨ oberg, On Stanley-Reisner rings, Topics in Algebra, Bana ch Center Publications, 26 (2) (1990), 57–70","cited_arxiv_id":null,"evidence_quote":"Fr\\\"oberg's characterization of edge ideals of regularity $2$, used in the sharpness discussion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows powers of cochordal edge ideals have linear resolutions, giving $\\operatorname{reg}(I(G)^s)=2s$ for complete bipartite graphs and hence equality in the bound."}],"review_version":1}