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arxiv: 2606.24426 · v1 · pith:V25NXU2Unew · submitted 2026-06-23 · 🧮 math.CO

Odd cycles in symmetric Cayley graphs on prime cyclic groups

Wei Li , Kai Yang This is my paper

Pith reviewed 2026-06-25 23:16 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C2505C3505C38
keywords Cayley graphsodd cyclesextremal graph theoryzero-sum problemspancyclicitycyclic groupssymmetric setsprime order
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0 comments X

The pith

The largest symmetric subset of Z_p avoiding a C_{2ℓ+1} in the Cayley graph has size exactly 2 floor((p + 2ℓ + 1)/(2(2ℓ + 1))) when p exceeds 2ℓ + 1, and zero otherwise.

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

The paper determines the precise maximum number of symmetric nonzero connections one can choose in the cyclic group of odd prime order p so that the resulting undirected Cayley graph contains no cycle of one prescribed odd length 2ℓ + 1. When the prime p itself equals 2ℓ + 1 the only possibility is the empty connection set. For larger primes the size is bounded by twice the floor of (p plus 2ℓ + 1) divided by twice the cycle length, and this bound is achieved. The argument relies on translating cycle-freeness questions into additive zero-sum conditions and then using a weak pancyclicity property to reduce the girth condition to the single forbidden length. A reader interested in vertex-transitive graphs or additive combinatorics therefore obtains an exact density threshold for a natural family of circulant graphs.

Core claim

Confirming a conjecture of Cashman and Kelley, the authors prove that ex_Cay(C_{2ℓ+1}, Z_p) equals zero when p equals 2ℓ + 1 and equals 2 floor((p + 2ℓ + 1)/(2(2ℓ + 1))) when p is larger. The proof combines a sharp additive zero-sum odd-girth argument with weak odd pancyclicity to move from control of the odd girth to control of one fixed odd cycle length. They also exhibit a canonical extremal family, give an exact extremality criterion phrased in terms of odd zero-sum avoidance, and supply an example showing that extremizers need not be dilates of the canonical construction.

What carries the argument

The sharp additive zero-sum odd-girth argument combined with weak odd pancyclicity, which converts exclusion of every odd cycle whose girth is 2ℓ + 1 into exclusion of the single cycle length 2ℓ + 1.

If this is right

  • A canonical family of symmetric sets achieves the bound for every pair p, ℓ.
  • Extremal sets are characterized exactly by the absence of odd zero-sums of length 2ℓ + 1.
  • Extremizers need not arise as dilates of the canonical construction.
  • The same numerical bound holds uniformly for all odd primes larger than the cycle length.

Where Pith is reading between the lines

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

  • The zero-sum criterion may supply analogous bounds for Cayley graphs on non-prime cyclic groups.
  • The transfer technique could be tested on other vertex-transitive graphs whose adjacency is defined by additive structure.
  • Connections to broader questions about the relationship between girth and pancyclicity in circulant graphs become visible once the zero-sum translation is in place.

Load-bearing premise

The combination of the sharp additive zero-sum odd-girth argument with weak odd pancyclicity correctly transfers exclusion of all odd cycles of girth 2ℓ + 1 into exclusion of the single cycle length 2ℓ + 1 for these Cayley graphs.

What would settle it

A symmetric set S whose cardinality exceeds the stated floor expression yet whose Cayley graph on Z_p still contains no cycle of length exactly 2ℓ + 1, or a nonempty symmetric S when p equals 2ℓ + 1 with the same property.

read the original abstract

Let $p$ be an odd prime and let $S\subseteq \Z_p$ be symmetric with $0\notin S$. Let $\Cay(\Z_p,S)$ be the undirected Cayley graph on $\Z_p$ in which $x$ and $y$ are adjacent if and only if $x-y\in S$. For $1\le \ell\le (p-1)/2$, define \[ \ex_{\Cay}(C_{2\ell+1},\Z_p)=\max\{|S|: S=-S,\ 0\notin S,\ \Cay(\Z_p,S)\text{ contains no }C_{2\ell+1}\}. \] Confirming a conjecture of Cashman and Kelley, we prove that if $p=2\ell+1$, then $\ex_{\Cay}(C_{2\ell+1},\Z_p)=0$, while if $p>2\ell+1$, then \[ \ex_{\Cay}(C_{2\ell+1},\Z_p)=2\floor{\frac{p+2\ell+1}{2(2\ell+1)}}. \] The proof combines a sharp additive zero-sum odd-girth argument with weak odd pancyclicity to transfer the result from odd-girth exclusion to fixed odd-cycle exclusion. We also give a canonical extremal family, an exact extremality criterion in terms of odd zero-sum avoidance, and an example showing that extremizers need not be dilates of the canonical construction.

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 paper proves that for odd prime p and 1 ≤ ℓ ≤ (p-1)/2 the extremal function ex_Cay(C_{2ℓ+1}, Z_p) equals 0 when p = 2ℓ+1 and equals 2⌊(p + 2ℓ + 1)/(2(2ℓ + 1))⌋ otherwise. This confirms the Cashman-Kelley conjecture. The argument combines a sharp additive zero-sum result controlling odd girth with weak odd pancyclicity to pass from girth exclusion to exclusion of the single cycle length 2ℓ+1; the manuscript also supplies a canonical extremal family, an exact zero-sum criterion for extremality, and an example of a non-dilate extremizer.

Significance. If the central transfer holds, the result supplies an exact, closed-form extremal function for a natural family of Cayley graphs together with an explicit construction and a non-dilate counter-example to uniqueness. The explicit odd-zero-sum avoidance criterion and the use of two independent combinatorial tools (additive combinatorics and pancyclicity) are concrete strengths that make the claim falsifiable and reproducible.

minor comments (2)
  1. [Abstract] The abstract states the range 1 ≤ ℓ ≤ (p-1)/2 but the formula is written without explicit qualification when p = 2ℓ + 1; a single sentence clarifying the two cases would prevent momentary confusion.
  2. The term 'weak odd pancyclicity' is used without a local definition or pointer to a standard reference; a one-sentence gloss in §2 would help readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main results, including the confirmation of the Cashman-Kelley conjecture, the combination of additive combinatorics and pancyclicity arguments, the canonical extremal family, the zero-sum criterion, and the non-dilate example.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper confirms an external conjecture of Cashman and Kelley via a direct combinatorial argument that combines a sharp additive zero-sum result on odd girth with weak odd pancyclicity to transfer girth exclusion to fixed-cycle exclusion. No equations reduce a claimed prediction to a fitted parameter by construction, no self-citation is load-bearing for the central theorem, and the extremal formula is derived from the stated tools rather than renamed or smuggled via prior author work. The argument is presented as independent of the target bound.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts about cyclic groups of prime order, symmetric subsets, and the definition of Cayley graphs; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Z_p forms an additive cyclic group of prime order
    Used as the vertex set and difference set throughout.
  • standard math Cayley graphs on abelian groups are vertex-transitive and undirected when S is symmetric
    Standard background invoked by the definition of ex_Cay.

pith-pipeline@v0.9.1-grok · 5795 in / 1396 out tokens · 27846 ms · 2026-06-25T23:16:22.799278+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Tur\'an Theorem for Cayley Graphs

    math.CO 2026-06 unverdicted novelty 6.0

    The Cayley-Turán number exCay(K_{r+1}, Z_p) for odd prime p equals p-1 - 2 floor(p/(r+1)).

Reference graph

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