Note on a conjecture of Graham

Area of Science:

• Combinatorics
• Group Theory
• Number Theory

Background:

• Graham's conjecture addresses zero-sum subsequences in cyclic groups.
• Previous proofs for this conjecture were complex, especially for smaller primes.
• Existing advanced methods were used for non-prime cases, but lacked simplicity.

Purpose of the Study:

• To provide a simple proof for Graham's conjecture on zero-sum subsequences.
• To extend the proof's validity to non-prime moduli.
• To detail the structure of sequences satisfying the conjecture's conditions.

Main Methods:

• Utilized the Cauchy-Davenport Theorem and the pigeonhole principle for a simple proof.
• Employed the Devos-Goddyn-Mohar Theorem for an alternative proof of the non-prime case.
• Extended the methodology to arbitrary finite abelian groups.

Main Results:

• A short, elementary proof of Graham's conjecture is presented.
• The proof is valid for both prime and non-prime moduli.
• An exhaustive structural characterization of the sequences is achieved.

Conclusions:

• The conjecture is proven with elementary tools, fulfilling a long-standing quest for simplicity.
• The methods provide a unified approach applicable to a broader range of groups.
• This work simplifies and extends existing results in additive combinatorics.