{"id":"ac71a57d-8319-4343-b87b-23fe4da79493","arxiv_id":"2507.05700","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For edge ideals of connected graphs, the v-number can be arbitrarily larger or smaller than the degree of the h-polynomial, their sum is at most the number of vertices, and equality holds exactly for disjoint unions of star graphs.","lead":"The paper proves that for any integer m, there is a connected graph whose v-number and degree of its h-polynomial differ by exactly m, and it classifies when their sum is as large as possible. It also shows, with explicit computer checks, that all 13 possible orderings among three algebraic invariants of edge ideals can occur.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6's v-number lower-bound proof asserts a false implication: in H(2,3), B={x2,z1} has N(B) minimal but avoids triangle 1, so the proof as written is invalid.","rationale":"The central claim, Theorem 1.1, splits into two directions: positive differences via Theorem 3.4 and negative differences via Theorem 3.6. The reader's weakest assumption concerned unshipped Macaulay2 data for G\\A in the positive direction. My stress-test found a different, more immediate problem in the negative direction: the proof of Theorem 3.6 contains a false implication. The example H(2,3) with B={x2,z1} shows that a minimizing independent set need not meet every triangle, so the written argument for v(I(G))≥v is incomplete. The theorem appears salvageable by a short case analysis, so this does not justify rejection. The computational reproducibility concerns identified by the reader are also valid and should be fixed, but the proof gap is the more load-bearing issue for the central theorem. The independent parts of the paper, such as Proposition 3.5, Lemma 3.2, and the bound in Theorem 4.2, look sound. Since the reader already returned CONDITIONAL and my concern does not change that assessment, the verdict is UNCHANGED.","tokens_in":20508,"tokens_out":28616,"duration_ms":322993,"concrete_test":"Enumerate all independent sets B of H(2,3) with N_G(B) a minimal vertex cover, either directly or with a short Macaulay2 script. If the minimum cardinality is 2, Theorem 3.6's conclusion survives and only the proof needs the missing case; if it is 1, the theorem is false. In the same enumeration, confirm that B={x2,z1} has N_G(B)={x1,w1,y2,w2} minimal and B∩T1=∅, which falsifies the intermediate claim in the proof. Optionally repeat the enumeration for H(3,4) to test the general lower-bound pattern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is in the proof of Theorem 3.6 (Section 3). After observing that a minimal vertex cover N_G(B) contains at most two vertices from each triangle {x_i,y_i,w_i}, the proof asserts: 'Consequently, {x_i,y_i,w_i}∩B ≠ ∅ for each 1≤i≤v.' This implication is false for i=1. In H(2,3) (Construction 3 with v=2, d=3), take T1={x1,y1,w1}, T2={x2,y2,w2}, the edge x1x2, and the leaf z1 attached to w1. The set B={x2,z1} is independent, and N_G(B)={x1,w1,y2,w2} is a minimal vertex cover of H(2,3), yet B∩T1=∅. The claimed lower bound |B|≥v therefore does not follow from the supplied argument; one must add a separate case: if B misses T1, then B must contain a leaf z_j, replacing the missing T1 vertex, since otherwise the edge y1w1 is uncovered. This matters because Theorem 3.6 supplies the entire 'deg(h)−v can be arbitrarily large' direction of Theorem 1.1. The theorem may still be true, but the proof as written contains a false intermediate claim. Secondary but real: Theorem 3.4 quotes Macaulay2 values for G\\A without shipping that computation, and Example 5.11 explicitly says its headline values could not be checked directly.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the relative sizes of the v-number v(I(G)), the degree d(G) of the h-polynomial of R/I(G), and the Castelnuovo--Mumford regularity for edge ideals of connected graphs. It claims Theorem 1.1: for every integer m there is a connected graph with v(I(G))-deg(h_{R/I(G)}(t))=m; Theorem 3.4 gives the positive direction by attaching n copies of an 11-vertex base graph (from Jaramillo--Villarreal) to a complete graph, and Theorem 3.6 constructs, for every 1<=v<=d, a connected graph H(v,d) with v(I)=v and deg(h)=d. Section 4 proves the sharp bound v(I(G))+deg(h_{R/I(G)}(t))<=n, with equality exactly for disjoint unions of stars. Section 5 exhibits all thirteen possible inequalities among v, deg(h), and regularity. The paper also reports a computer search showing that the base graph is one of two minimal examples on 11 vertices and 25 edges with v>deg(h), and that no smaller example exists.","tokens_in":20830,"tokens_out":10220,"duration_ms":107165,"significance":"If the results hold, the paper resolves the natural comparison problem between the v-number and the degree of the h-polynomial for connected edge ideals, complementing the known results for regularity by Hibi--Matsuda--Van Tuyl and by Biermann et al. The construction lemmas (Lemmas 3.2 and 3.3) are potentially reusable tools for building graphs with prescribed invariants. The paper ships Macaulay2 code for the two base graphs and includes an exhaustive computer search for minimal counterexamples, which are positive features. However, the proof of the arbitrarily-large-deg(h) direction (Theorem 3.6) contains a false inference, and several load-bearing Macaulay2 computations (for the induced subgraph G\\A and for the exhaustive search) are not archived. These gaps currently prevent the paper from being fully verified in its present form.","major_comments":[{"comment":"The proof of the lower bound v(I(G)) >= v contains a false inference. After showing that a minimal vertex cover NG(B) contains at most two vertices from each triangle Ti={xi,yi,wi}, the proof asserts 'Consequently, {xi,yi,wi} intersect B is nonempty for each 1<=i<=v.' This is false. In H(2,3) (Construction 3 with v=2, d=3), take B={x2,z1}. Then B is independent and NG(B)={x1,w1,y2,w2} is a minimal vertex cover of H(2,3), yet B intersect T1 is empty. Hence the claimed lower bound |B|>=v does not follow from the preceding sentence. A repair is available: if B misses T1, then to cover the edge y1w1, B must either contain a vertex of T1 or contain a leaf z_j, and since each leaf is adjacent only to w1, one can still obtain the bound |B|>=v by a case analysis. The theorem may be true, but the proof as written is incomplete and needs this additional case.","section":"Section 3, proof of Theorem 3.6"},{"comment":"The proof relies on Macaulay2 computations for the induced subgraph G\\A with A={x1,...,x5}: alpha(G\\A)=2, deg(h_{R/I(G\\A)})=2, v(I(G\\A))=2, and reduced Hilbert series (1+4t+t^2)/(1-t)^2. The appendix contains code for the base graph G but not for G\\A or for v(I(G\\A)). These data are load-bearing: they are precisely the hypotheses needed to apply Lemmas 3.2 and 3.3 and to conclude v(I(H_n))=3n and deg(h_{R/I(H_n)})=2n+1. Please include the code (or an explicit edge list for G\\A and the corresponding commands) so that this step is independently checkable.","section":"Section 3, proof of Theorem 3.4"},{"comment":"The exhaustive search over all graphs with at most 11 vertices and 25 edges is stated to involve well over 100,000,000 graphs, but no script or detailed description of the search and verification is provided. As a computational theorem, this claim is not independently verifiable from the manuscript as it stands. Since the paper already ships Macaulay2 code elsewhere, please provide the search code or a repository link, or at least a precise description of the filtering and checking procedure.","section":"Section 3, Theorem 3.1"}],"minor_comments":[{"comment":"The text first states 'using Macaulay2, one can check that d=8' and later says 'We were not able to verify these values directly using Macaulay2 since the computations would not finish.' Please clarify which values were verified computationally and supply a proof or code for d=8.","section":"Example 5.11"},{"comment":"Several examples (e.g., Examples 5.3, 5.5, 5.6, 5.8, 5.9) quote values of d and v obtained with Macaulay2 without shipping the corresponding code. A supplementary file with all such computations would improve reproducibility.","section":"Section 5"},{"comment":"The caption does not label the horizontal and vertical axes; please label them as v(I) and deg(h), and state whether the observed inequality 2v(I(G))+deg(h) <= n+1 is a proved result or an empirical observation.","section":"Appendix A, Figure 11"},{"comment":"There is a typo: 'let us proof' should be 'let us prove'.","section":"Example 5.11, proof of claim"}],"recommendation":"major_revision","confidential_remarks":"The false implication in Theorem 3.6 is the main obstacle: it is a genuine error in a central proof, but it appears repairable by a short case analysis. The unarchived Macaulay2 data for G\\A and for the exhaustive search are also notable reproducibility gaps in a computationally heavy paper. The remaining structural results (Lemma 3.2, Lemma 3.3, Theorem 4.2, and the constructions in Section 5) appear coherent. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a worthwhile paper—the comparison question is natural, the constructions are explicit, and the bound/classification in Theorem 4.2 is clean—but one of the two main proofs has a genuine gap, and the computational claims are under-documented.\n\nWhat's new: prior work compared v with regularity and regularity with deg h, but not v with deg h for edge ideals. The paper closes that gap. Theorem 1.1 (arbitrary difference), the realizability of every pair (v,d) with 1 ≤ v ≤ d by a connected graph, and the sharp bound v + deg h ≤ n with equality iff disjoint union of stars are all credible and useful. The 13 orderings in Section 5, including characteristic dependence, are a nice concrete addition. The base example from Jaramillo-Villarreal is well used, and the gluing lemmas may be reusable beyond this paper.\n\nWhere I'd push back: the proof of Theorem 3.6 contains a false implication. The stress-test example is right: in H(2,3), B = {x2,z1} is independent with N_G(B) = {x1,w1,y2,w2} a minimal vertex cover, yet B misses triangle 1 entirely. So the claim that a minimal cover containing at most two vertices from each triangle forces B to meet every triangle is not valid. The lower bound v(I(G)) ≥ v may still be true—you likely need a separate case using the leaves—but as written the proof does not establish it. This is not a minor typo; it is the argument for the entire \"degree can exceed v arbitrarily\" direction.\n\nTwo smaller issues. Theorem 3.1 is an exhaustive computer search over 86 million graphs with no search code included; the appendix only has code for the two base graphs. Example 5.11 quotes d = 8 in a way that is hard to reconcile with the sentence that computations would not finish. These are reproducibility gaps, not evidence of error, but they should be fixed.\n\nNet: the results are probably right, the paper is written carefully overall, and a good referee can get it into shape. Commutative algebraists working on edge ideals and their invariants will want this. Send to review, but require the Theorem 3.6 proof to be repaired and the computational claims to be backed by code.","headline":"Solid and publishable in principle, but the proof of Theorem 3.6 has a false step that needs fixing, and the computational claims need code.","tokens_in":21355,"tokens_out":3853,"would_cite":true,"duration_ms":42914,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E40","13D02","13D40","13F55","05C69"],"pacs":[],"model":"deepseek-v4-flash","headline":"The $v$-number of an edge ideal and the degree of its $h$-polynomial can be tuned independently on connected graphs: every integer difference, and in fact every pair $(v,d)$ with $1 \\leq v \\leq d$, is realized by some explicit graph.","keywords":["v-number","degree of h-polynomial","Castelnuovo-Mumford regularity","edge ideal of a graph","connected graphs","star graphs","Hilbert series"],"falsifier":"Compute the $v$-number, $h$-polynomial degree, independence number, and reduced Hilbert series of the induced subgraph $G\\setminus A$ for the 11-vertex graph $G$ in Figure 1, where $A=\\{x_1,\\ldots,x_5\\}$; if any quoted value differs from $\\alpha(G\\setminus A)=2$, $\\deg(h_{R/I(G\\setminus A)}(t))=2$, $v(I(G\\setminus A))=2$, or $(1+4t+t^2)/(1-t)^2$, then Theorem 3.4's construction no longer produces connected graphs with $v(I)-\\deg(h)=n$, although the $m\\leq 0$ direction of Theorem 3.6 would still hold.","tokens_in":20262,"feed_emoji":"","tokens_out":14285,"duration_ms":137369,"temperature":0.7,"pith_summary":"A graph's edge ideal has several numerical invariants; this paper compares two of them: the $v$-number and the degree of the $h$-polynomial. On connected graphs these two numbers are shown to be adjustable independently: for every integer $m$ there is a connected graph $G$ with $v(I(G))-\\deg(h_{R/I(G)}(t))=m$, and in fact every pair $(v,d)$ with $1 \\leq v \\leq d$ occurs as the two invariants of some connected graph. The paper also proves a sharp cap: for any graph on $n$ vertices, $v(I(G))+\\deg(h_{R/I(G)}(t)) \\leq n$, with equality exactly for disjoint unions of star graphs. These results give a way to control the two invariants separately and show that the $v$-number carries information beyond the Hilbert series and the regularity.","feed_headline":"Two graph invariants can be made to differ by any integer","feed_subtitle":"Explicit connected graphs realize every pair (v,d) with 1 ≤ v ≤ d, and v + d never exceeds the number of vertices.","key_machinery":"The argument runs through three pieces. The $v$-number of an edge ideal is read combinatorially: $v(I(G))$ is the minimum size of an independent set $A$ whose neighbour set $N_G(A)$ is a minimal vertex cover, so $v(I(G))=\\min\\{|A| \\mid A \\in \\mathcal{A}_G\\}$. A gluing lemma (Construction 1 and Lemma 3.2) assembles $n$ graphs $G_i$ into one connected graph $H_n$ by adding new vertices $y_i$, joining $y_i$ to a chosen subset $A_i$ of $G_i$, and making the $y_i$'s a clique; under the hypotheses that $\\alpha(G_i)-\\alpha(G_i\\setminus A_i)$ is odd and $\\deg(G_i)-\\deg(G_i\\setminus A_i)=\\alpha(G_i)-\\alpha(G_i\\setminus A_i)-1$ with matching signs of leading $h$-coefficients, it gives $\\deg(h_{R/I(H_n)}(t))=1+\\sum_i\\deg(h_{R_i/I(G_i)}(t))$, and under $v(I(G_i)) \\geq 1+v(I(G_i\\setminus A_i))$ it gives $v(I(H_n))=\\min_i[1+v(I(G_i\\setminus A_i))+\\sum_{j\\neq i}v(I(G_j))]$. The base graph $G$ of Figure 1 satisfies these hypotheses for $A=\\{x_1,\\ldots,x_5\\}$, giving $v=3>\\deg(h)=2$, and iterating the gluing makes the difference arbitrarily large. The final piece is the Hilbert-series formula for edge ideals, which implies $\\deg(h_{R/I(G)}(t)) \\leq \\alpha(G)$; together with $v(I(G)) \\leq \\beta(G)$ this yields the sum bound $v+\\deg(h) \\leq n$ and identifies equality with disjoint unions of star graphs.","core_discovery":"The paper's central result is Theorem 1.1: for every integer $m$ there is a connected graph $G$ with $v(I(G))-\\deg(h_{R/I(G)}(t))=m$. The direction $m>0$ is obtained by gluing $n+1$ copies of an 11-vertex, 25-edge base graph with $v=3$ and degree $2$: adding a new vertex per copy that is joined to five fixed vertices of that copy and connecting all new vertices in a clique makes $v=3(n+1)$ and $\\deg(h)=2n+3$. The direction $m \\leq 0$ is obtained from an explicit family $H(v,d)$ built from $v$ triangles, a star on the first $v$ vertices, and $d-v$ pendant leaves attached to one triangle vertex; a short-exact-sequence calculation gives $v(I(H(v,d)))=v$ and $\\deg(h_{R/I(H(v,d))}(t))=d$ for every $1 \\leq v \\leq d$. The paper also proves the sharp bound $v(I(G))+\\deg(h_{R/I(G)}(t)) \\leq n$ and classifies equality as exactly the disjoint unions of star graphs, and a computer search shows the base graph is minimal: no connected graph with fewer vertices, or with 11 vertices and fewer than 25 edges, has $v(I(G))>\\deg(h_{R/I(G)}(t))$, while at (11,25) there are exactly two such graphs.","pith_inferences":["The appendix tables for connected graphs up to 10 vertices suggest the sharper inequality $2v(I(G))+\\deg(h_{R/I(G)}(t)) \\leq n+1$, which the paper does not assert; proving it would refine Theorem 4.2.","The gluing lemma is written for any base graphs satisfying its parity, sign, and $v$-number hypotheses, so other base cases besides the 11-vertex example could produce families with different growth rates of $v(I)-\\deg(h)$; the paper only explores the one base graph.","An independent check of the quoted invariants of $G\\setminus A$ would settle the main numerical input of Theorem 3.4, since the supplied code covers the base graph but not this auxiliary subgraph."],"forward_implications":["For every positive integer $n$, the construction yields a connected graph with $v(I)=3(n+1)$ and $\\deg(h)=2n+3$, so the $v$-number can be made arbitrarily larger than the $h$-degree.","For every pair $(v,d)$ with $1 \\leq v \\leq d$, the family $H(v,d)$ realizes exactly that pair; in particular the $h$-degree can be made arbitrarily larger than the $v$-number while keeping $v=1$.","On any graph with $n$ vertices, $v(I(G)) + \\deg(h_{R/I(G)}(t)) \\leq n$, and the only graphs attaining equality are disjoint unions of star graphs.","All thirteen possible inequality patterns among $v(I(G))$, $\\deg(h_{R/I(G)}(t))$, and the Castelnuovo-Mumford regularity occur for edge ideals of connected graphs; some examples depend on the field characteristic through the regularity, while $v$ and the $h$-degree are characteristic-free.","No connected graph on at most 10 vertices, and no connected graph on 11 vertices with at most 24 edges, has $v(I(G)) > \\deg(h_{R/I(G)}(t))$; the two 11-vertex, 25-edge graphs in Figures 1 and 2 are the unique minimal exceptions."],"supporting_citations":[{"why":"Supplies the 11-vertex base graph with v=3 > deg(h)=2, and the independent-set description of the v-number used throughout.","marker":"[28]"},{"why":"Provides the v-number facts v(I(G)) ≤ β(G) and v(I(G):x)=v(I(G\\setminus N_G[x])) used in Lemmas 3.3 and 4.1.","marker":"[35]"},{"why":"Gives the additivity of regularity, Hilbert series, h-degree, and v-number for ideals in disjoint variable sets used in Lemma 2.3.","marker":"[27]"},{"why":"Establishes the comparable bound reg(I(G)) + deg(h) ≤ n that Theorem 4.2 mirrors.","marker":"[26]"},{"why":"Provides the independent-set Hilbert-series formula that yields deg(h) ≤ α(G) in Theorem 4.2.","marker":"[36]"},{"why":"Shows the v-number can be arbitrarily larger than the regularity, the construction problem adapted here to the h-degree.","marker":"[8]"}],"fun_headline_variants":["Set v-number minus h-degree to any integer","Every pair (v,d) with v≤d realizable by a connected graph","v + h-degree ≤ n; equality exactly for disjoint unions of stars","Exactly two 11-vertex, 25-edge graphs are minimal with v > h°"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the computer-algebra values quoted for the auxiliary induced subgraph $G\\setminus A$ of the 11-vertex base graph are correct: $\\alpha(G\\setminus A)=2$, $\\deg(h_{R/I(G\\setminus A)}(t))=2$, $v(I(G\\setminus A))=2$, and reduced Hilbert series $(1+4t+t^2)/(1-t)^2$; the paper supplies code for the base graph $G$ but not for $G\\setminus A$, and Theorem 3.4 uses these values to verify the hypotheses of Lemma 3.2 and Lemma 3.3.","fun_headline_variants_meta":{"raw":{"variants":["Set v-number minus h-degree to any integer","Every pair (v,d) with v≤d realizable by a connected graph","v + h-degree ≤ n; equality exactly for disjoint unions of stars","Exactly two 11-vertex, 25-edge graphs are minimal with v > h°"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000503,"raw_usage":{"total_tokens":2564,"prompt_tokens":1156,"completion_tokens":1408,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":1328}},"tokens_in":772,"tokens_out":1408,"duration_ms":15969,"temperature":1.0,"reasoning_tokens":1328,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:23:19.507201+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $v$-number, $h$-polynomial degree, independence number, and reduced Hilbert series of the induced subgraph $G\\setminus A$ for the 11-vertex graph $G$ in Figure 1, where $A=\\{x_1,\\ldots,x_5\\}$; if any quoted value differs from $\\alpha(G\\setminus A)=2$, $\\deg(h_{R/I(G\\setminus A)}(t))=2$, $v(I(G\\setminus A))=2$, or $(1+4t+t^2)/(1-t)^2$, then Theorem 3.4's construction no longer produces connected graphs with $v(I)-\\deg(h)=n$, although the $m\\leq 0$ direction of Theorem 3.6 would still hold.","supporting_citations":[{"cited_title":"Villarreal, The v-number of edge ideals , J","cited_arxiv_id":null,"evidence_quote":"Supplies the 11-vertex base graph with v=3 > deg(h)=2, and the independent-set description of the v-number used throughout."},{"cited_title":"Algebraic Combin","cited_arxiv_id":null,"evidence_quote":"Provides the v-number facts v(I(G)) ≤ β(G) and v(I(G):x)=v(I(G\\setminus N_G[x])) used in Lemmas 3.3 and 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the additivity of regularity, Hilbert series, h-degree, and v-number for ideals in disjoint variable sets used in Lemma 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the comparable bound reg(I(G)) + deg(h) ≤ n that Theorem 4.2 mirrors."},{"cited_title":"Stanley, Combinatorics and commutative algebra , Second, Progress in Mathematics, vol","cited_arxiv_id":null,"evidence_quote":"Provides the independent-set Hilbert-series formula that yields deg(h) ≤ α(G) in Theorem 4.2."},{"cited_title":"Algebraic Combin","cited_arxiv_id":null,"evidence_quote":"Shows the v-number can be arbitrarily larger than the regularity, the construction problem adapted here to the h-degree."}],"review_version":1}