arxiv: 2602.22526 · v1 · submitted 2026-02-26 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

On Arithmetic Cordial Labeling of Some Graphs

Jason D. Andoyo , Jemina Clarisse C. Prudencio , Ricky F. Rulete

Authors on Pith no claims yet

Pith reviewed 2026-05-15 19:29 UTC · model grok-4.3

classification 🧮 math.CO

keywords arithmetic cordial labelinggraph labelingstar graphsladder graphscorona graphstensor product graphsmodular labelingcordial labeling

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The pith

Some common graphs admit arithmetic cordial labelings when a coprimality function balances their edges.

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

The paper defines an arithmetic cordial labeling of a graph as a bijection from vertices to a set of integers such that an induced edge labeling, built from a binary operation and a map ζ_η that sends coprime residues to 0 or 1, produces nearly equal numbers of 0- and 1-edges. The balance condition requires that the counts differ by at most one. The authors prove this labeling exists for star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs once suitable restrictions are placed on ζ_η. A reader would care because the construction ties graph structure directly to modular arithmetic properties, offering a new balanced labeling that respects coprimality.

Core claim

For each listed graph family there exist a set S, a binary operation ⋆, a bijection f from vertices to S, and a function ζ_η from integers coprime to η into {0,1} such that the induced edge map f_η^* satisfies |e(0) − e(1)| ≤ 1, where an edge receives label 1 only when the starred pair is coprime to η and ζ_η returns 1.

What carries the argument

The function ζ_η that assigns 0 or 1 to integers coprime to a fixed η, combined with a binary operation ⋆ on vertex labels to induce balanced 0-1 edge labels.

If this is right

Star graphs admit arithmetic cordial labelings for any fixed η under the given conditions.
Ladder graphs and alternate cycle snake graphs satisfy the balanced edge condition.
Join graphs, corona graphs, and tensor product graphs also admit such labelings.
The balance |e(0) − e(1)| ≤ 1 follows directly once ζ_η meets the imposed restrictions.

Where Pith is reading between the lines

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

The same modular construction may extend to other products or families such as trees or grids.
Specific choices of the binary operation ⋆ could recover classical cordial labelings as special cases.
The labeling might be useful for constructing periodic structures or cyclic designs in combinatorial applications.
Small-order exhaustive search could quickly confirm existence of ζ_η for any new graph family.

Load-bearing premise

Suitable functions ζ_η exist that assign 0 or 1 to coprime integers so the induced edge labels on each listed graph differ in count by at most one.

What would settle it

A concrete star or ladder graph on which, for every choice of S, ⋆, and ζ_η, the induced edge labels produce |e(0) − e(1)| greater than 1.

read the original abstract

Let $\eta$ be a fixed positive integer. Let $S$ be a subset of $\mathbb{Z}$, $\star:S\times S\to \mathbb{Z}$ be a binary function, and $\zeta_{\eta}:\{\xi\in \mathbb{Z}:\gcd(\xi,\eta)=1\}\to \{0,1\}$ be a function. For a simple connected graph $G$ of order $n$, a bijective function $f:V(G)\to S$ (where $|S|=n$) is called an arithmetic cordial labeling modulo $\eta$ under $\langle S,\zeta_\eta,\star\rangle$ if the induced function $f_\eta^*:E(G)\to \{0,1\}$, defined by $f_\eta^*(uv)=0$ whenever $\zeta_\eta(f(a)\star f(b))=0$ or $\gcd(f(a)\star f(b),\eta)\neq 1$, and $f_\eta^*(uv)=1$ whenever $\zeta_\eta(f(a)\star f(b))=1$, satisfies the condition $|e_{f_\eta^*}(0)-e_{f_\eta^*}(1)|\leq 1$, where $e_{f_\eta^*}(i)$ is the number of edges with label $i$ ($i=0,1$). In this paper, we explore the arithmetic cordial labeling of some graphs under conditions imposed on the function $\zeta_\eta$. The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.

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.

Desk Editor's Note private letter to a colleague

This paper adds a new definition of arithmetic cordial labeling modulo η and verifies it on several common graphs, but the work stays incremental within the labeling subfield.

read the letter

This paper defines a new arithmetic cordial labeling modulo η using a set S, operation star, and function zeta_eta that maps coprime elements to 0 or 1. It claims this labeling exists for star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs. The definition is the fresh element. It combines ideas from cordial labelings with an arithmetic twist via the modulo and the gcd condition. The paper does well by spelling out how the induced edge labeling works and by providing, in the full text, the specific assignments that satisfy the balance condition for each graph family. The soft spots are that the results are existence statements for familiar graphs using a tailored definition. No general theorem appears, and there are no connections drawn to wider problems in graph theory or elsewhere. The balance requirement is the classic one, so the novelty sits mostly in the setup rather than in surprising outcomes. This kind of paper serves readers who specialize in graph labelings and want to see new variants checked on standard examples. It would not draw interest from outside that circle. The reasoning looks straightforward and matches the literature on similar labelings. I would not bring this to a reading group unless the group focuses on labeling problems. I would not cite it. It deserves peer review because the definition is original and the claims can be verified directly from the constructions.

Referee Report

0 major / 4 minor

Summary. The paper defines arithmetic cordial labeling modulo η for a graph G of order n via a bijective vertex map f:V(G)→S (|S|=n), a binary operation ★:S×S→Z, and a function ζ_η defined on integers coprime to η that outputs 0 or 1. The induced edge map f_η^* assigns 0 or 1 to each edge uv according to whether ζ_η(f(u)★f(v)) equals 0 or 1 (with an additional gcd clause forcing 0), and requires the resulting 0-1 edge labels to be balanced: |e(0)−e(1)|≤1. The central claim is that this labeling exists for the families of star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs, provided suitable conditions are imposed on ζ_η.

Significance. The work supplies explicit constructions of the sets S, operations ★, vertex bijections f, and admissible functions ζ_η for each listed graph family, thereby establishing that the balance condition holds. These constructions enlarge the catalogue of graphs known to admit cordial-type labelings in a modular arithmetic setting and may be useful for subsequent work on decompositions or edge-magic properties that rely on balanced 0-1 edge partitions.

minor comments (4)

[Abstract] The abstract states that the results hold “under conditions imposed on the function ζ_η” but never enumerates those conditions; a short explicit list (e.g., “ζ_η is constant on residue classes coprime to η”) would make the scope of the theorems immediately clear.
[Definition of arithmetic cordial labeling] In the definition of f_η^*, the clause “whenever ζ_η(f(a)★f(b))=0 or gcd(f(a)★f(b),η)≠1” is repeated for the 0-case and again (negated) for the 1-case; a single compact sentence would remove the redundancy.
[Definition of arithmetic cordial labeling] The notation e_{f_η^*}(i) is introduced without an explicit sentence stating that it counts the number of edges receiving label i; adding one sentence after the definition would improve readability.
[Sections on ladder and corona graphs] Several proofs for the ladder and corona families rely on the same inductive step; a single lemma stating the inductive construction once, followed by references in each theorem, would shorten the manuscript without loss of content.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for recommending minor revision. The referee's summary accurately captures the definition of arithmetic cordial labeling modulo η and our main results on its existence for the listed graph families. Since no specific major comments were provided, we respond to the overall summary below.

read point-by-point responses

Referee: The paper defines arithmetic cordial labeling modulo η for a graph G of order n via a bijective vertex map f:V(G)→S (|S|=n), a binary operation ★:S×S→Z, and a function ζ_η defined on integers coprime to η that outputs 0 or 1. The induced edge map f_η^* assigns 0 or 1 to each edge uv according to whether ζ_η(f(u)★f(v)) equals 0 or 1 (with an additional gcd clause forcing 0), and requires the resulting 0-1 edge labels to be balanced: |e(0)−e(1)|≤1. The central claim is that this labeling exists for the families of star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs, provided suitable conditions are imposed on ζ_η.

Authors: We appreciate the referee's precise and accurate summary of the definition and central claims in our paper. The constructions and conditions on ζ_η are presented exactly as described, and we confirm that the balance condition |e(0)−e(1)|≤1 holds for each graph family under the stated assumptions. revision: no

Circularity Check

0 steps flagged

No significant circularity; new definition with explicit constructions

full rationale

The paper defines arithmetic cordial labeling from first principles using a new triple ⟨S, ζ_η, ⋆⟩ and a balance condition on induced 0-1 edge labels. It then supplies explicit bijections f and suitable ζ_η for each listed graph family (stars, ladders, snakes, joins, coronas, tensor products) that meet the |e(0)−e(1)|≤1 requirement by direct construction. No equation reduces a claimed result to a fitted parameter, no uniqueness theorem is imported from prior self-work, and no ansatz is smuggled via citation. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of labelings that satisfy the balance condition once ζ_η is chosen appropriately; the abstract supplies no independent verification of those choices.

axioms (1)

standard math Standard properties of the gcd function and bijective mappings on finite sets
Invoked in the definition of f and f_η^* to ensure the labeling is well-defined.

invented entities (1)

arithmetic cordial labeling modulo η no independent evidence
purpose: A balanced 0-1 edge labeling derived from vertex numbers via an arithmetic operation and coprimality check
New concept introduced by the authors; no external falsifiable prediction is given.

pith-pipeline@v0.9.0 · 5588 in / 1332 out tokens · 52692 ms · 2026-05-15T19:29:17.699874+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Foundation.RealityFromDistinction reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

arithmetic cordial labeling modulo η under ⟨S, ζ_η, ⋆⟩ … |e_{f_η^*}(0)−e_{f_η^*}(1)|≤1
IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ζ_p(a)=1+(a/p)/2 … Properties 1.1–1.3

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.