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arxiv: 2607.01438 · v1 · pith:XWEESIPYnew · submitted 2026-07-01 · 🧮 math.CO · cs.DM

Annihilation, Independence, and Residue: Sharp Matching Bounds for the Annihilation Gap and a TxGraffiti Application

Ohr Kadrawi , Vadim E. Levit This is my paper

Pith reviewed 2026-07-03 20:04 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords annihilation numberindependence numbermatching numberresidueannihilating decompositionTxGraffitiCaro-Wei boundKönig-Egerváry graph
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The pith

The gap between a graph's annihilation number and independence number is at most 2μ + 1 minus the ceiling of the square root of 6μ, where μ is the matching number, and this bound is tight for every value of μ.

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

The paper proves an exact upper bound on the difference a(G) minus α(G) that depends only on the size of a maximum matching. The bound is achieved by explicit constructions for every possible matching number at least one. It also gives matching-dependent sharp bounds for forests, bipartite graphs, and König-Egerváry graphs. Using annihilating decompositions and the classical residue bound, the paper proves a TxGraffiti-generated inequality relating annihilation number, residue, and maximum degree, and refines the Caro-Wei estimate with an explicit nonnegative slack term.

Core claim

For any finite simple graph G with matching number μ(G) at least 1, a(G) − α(G) ≤ 2μ(G) + 1 − ⌈√(6μ(G))⌉, and the bound is attained for every prescribed matching number. Sharp matching-dependent bounds hold for forests, bipartite graphs, and König-Egerváry graphs with explicit equality constructions and criteria. The annihilation-residue inequality α(G) ≥ (a(G) + res(G)) / Δ(G) holds for every connected graph of order at least three, both hypotheses are necessary, and a refined Caro-Wei bound with nonnegative slack is obtained whose equality cases are characterized in degree-sequence form.

What carries the argument

The closed-form bound a(G) − α(G) ≤ 2μ(G) + 1 − ⌈√(6μ(G))⌉ derived via annihilating decompositions together with the residue inequality res(G) ≤ α(G).

If this is right

  • Equality holds in the general bound for graphs realizing every integer value of the matching number.
  • The same bound is sharp when restricted to forests, to bipartite graphs, and to König-Egerváry graphs.
  • The TxGraffiti inequality α(G) ≥ (a(G) + res(G)) / Δ(G) is valid for every connected graph on at least three vertices.
  • A computable bracket for α(G) is obtained by combining the exact matching bound with the refined Caro-Wei estimate.
  • A Gupta-type residue bound for the annihilation gap follows directly from the matching formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bound supplies a fast, matching-based certificate that can be checked without computing α(G) itself.
  • The explicit slack term in the refined Caro-Wei estimate may tighten existing approximation algorithms that rely on degree sequences alone.
  • Equality criteria in degree-sequence form could be used to classify all graphs attaining the combined bracket.

Load-bearing premise

The classical result that the residue of any graph is at most its independence number holds and combines with annihilating decompositions to control the gap.

What would settle it

A single graph G with μ(G) ≥ 1 for which a(G) − α(G) exceeds 2μ(G) + 1 − ⌈√(6μ(G))⌉.

Figures

Figures reproduced from arXiv: 2607.01438 by Ohr Kadrawi, Vadim E. Levit.

Figure 1
Figure 1. Figure 1: The tree Ts used in Proposition 3.4: one arm has length 1, and the remaining s − 1 arms have length 2. Corollary 3.5. If F is a forest with at least one edge, then a(F) ≤ 3α(F) − 1 2 . Proof. By Theorem 3.1 and Lemma 2.4, a(F) − α(F) ≤ µ(F) − 1 2 ≤ α(F) − 1 2 . Rearranging gives the desired inequality. 4 Bipartite graphs The bound in this section holds for every bipartite graph. The non-tree hypothesis ent… view at source ↗
Figure 2
Figure 2. Figure 2: The six-vertex connected bipartite equality graph, with bipartition [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The connected bipartite equality family Hr. The core is Kr,r; every core vertex has one pendant leaf, and there are r 2 additional branches x1bib ′ i . 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

Let $G$ be a finite simple graph. The annihilation number $a(G)$ is an efficiently computable upper bound on the independence number $\alpha(G)$. We develop a sharp matching-number theory for the gap $a(G)-\alpha(G)$. The strongest general theorem is the exact closed form \[a(G)-\alpha(G)\leq 2\mu(G)+1- \lceil \sqrt{6 \mu(G)} \rceil \qquad(\mu(G)\geq 1), \] and the bound is attained for every prescribed matching number. We also prove sharp matching-dependent bounds for forests, bipartite graphs, and K\"onig-Egerv\'ary graphs, with equality constructions, equality certificates, and equality criteria. Finally, we treat a TxGraffiti output as a machine-conjecture case study. Using annihilating decompositions together with the classical Havel-Hakimi residue inequality $res(G)\leq \alpha(G)$, we give an independent proof of the TxGraffiti annihilation-residue inequality \[ \alpha(G)\geq \frac{a(G)+res(G)}{\Delta(G)} \] for every connected graph $G$ of order at least three, show that both hypotheses are necessary, and compare this proof with a recent Caro-Wei approach. We also refine the Caro-Wei annihilation estimate by an explicit nonnegative slack term, identify its equality cases in degree-sequence form, and combine the refinement with our exact matching-number bound to obtain a combined computable bracket for the independence number and a Gupta-residue bound for the annihilation gap.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops sharp matching-number bounds on the annihilation gap a(G)−α(G). The central theorem states that a(G)−α(G)≤2μ(G)+1−⌈√(6μ(G))⌉ for every graph with μ(G)≥1 and that the bound is attained for every prescribed integer value of μ(G). Separate sharp bounds, equality constructions, and equality criteria are given for forests, bipartite graphs, and König-Egerváry graphs. The paper also supplies an independent proof, via annihilating decompositions and the classical Havel-Hakimi residue inequality, of the TxGraffiti conjecture α(G)≥(a(G)+res(G))/Δ(G) for connected graphs of order at least three, shows both hypotheses are necessary, refines the Caro-Wei annihilation estimate by an explicit nonnegative slack term, and combines the new matching bound with the refinement to obtain computable brackets for α(G).

Significance. If the claimed bounds and proofs hold, the work supplies the first exact closed-form theory of the annihilation gap in terms of matching number, together with fully characterized equality cases. The independent, non-circular proof of the machine-generated annihilation-residue inequality and the explicit slack refinement of Caro-Wei are concrete strengths. The resulting computable lower and upper estimates for α(G) are likely to be of interest in extremal graph theory and in algorithmic applications of the annihilation number.

minor comments (2)
  1. [§1] §1, paragraph following Definition 1.2: the sentence defining the annihilation number a(G) via the maximum size of an annihilating set could be restated in one line for readers who encounter the parameter for the first time.
  2. [Theorem 3.4] Theorem 3.4 (equality criterion for forests): the statement lists three conditions but the proof only verifies the forward direction explicitly; adding a one-sentence remark that the converse follows from the construction in §3.2 would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept. The summary accurately captures the main results on the annihilation gap, the matching-number bounds, the independent proof of the TxGraffiti conjecture, and the Caro-Wei refinement.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results on the annihilation gap bound and the TxGraffiti inequality are derived via independent proofs that invoke the classical Havel-Hakimi residue inequality res(G) ≤ α(G), annihilating decompositions, and explicit equality constructions for every μ(G) ≥ 1. No equation or step reduces a claimed prediction or bound to a fitted parameter, self-definition, or load-bearing self-citation by construction; the TxGraffiti case study is explicitly presented as an independent verification rather than an assumption. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces no new free parameters, invented entities, or ad-hoc axioms. It rests on standard definitions of a(G), α(G), μ(G), res(G), and Δ(G) from the graph-theory literature and on the classical Havel-Hakimi inequality res(G)≤α(G).

axioms (2)
  • standard math Standard definitions and known properties of the annihilation number a(G), independence number α(G), matching number μ(G), residue res(G), and maximum degree Δ(G) for finite simple graphs.
    Invoked in the opening sentence and throughout the abstract as the foundation for the gap analysis and the residue inequality.
  • standard math The classical Havel-Hakimi residue inequality res(G)≤α(G).
    Explicitly used together with annihilating decompositions in the proof of the annihilation-residue inequality.

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