On zero-sum Ramsey numbers of cycles and wheels

TL;DR

This paper determines bounds and exact values for zero-sum Ramsey numbers of cycles and wheels in edge-labeled complete graphs over finite cyclic groups, extending classical combinatorial theorems.

Contribution

It introduces new bounds and exact formulas for zero-sum Ramsey numbers of cycles and wheels, utilizing insertion arguments rooted in the Erdős-Ginzburg-Ziv theorem.

Findings

Exact value for odd q ≥ 3: R(C_{qk}, Z_q) = qk + q - 1 for large k.

For q=3, R(C_{3k}, Z_3) = 3k + 2 for all k ≥ 2.

Resolved zero-sum Ramsey numbers for wheel graphs W_m = C_m + K_1 for m divisible by 3.

Abstract

For an integer $q\ge 2$ and a graph $F$ with $q\mid e(F)$, let $R(F,\Z_q)$ be the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\Z_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},\Z_q)\le \max\{R(C_{2q},\Z_q),qk+q-1\}$ via an insertion argument rooted in the classic Erd\H{o}s-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},\Z_q)\le \max\{35q^2,qk+q-1\}$ for every $q\ge 3$. We also show that $R(C_{qk},\Z_q)\ge qk+q-1$ for odd $q\ge 3$. Hence, for every fixed odd $q\ge 3$ and every $k\ge 35q$, we obtain the exact value $R(C_{qk},\Z_q)=qk+q-1$. For even $q\ge 4$, the same method gives $qk+\frac q2-1\le R(C_{qk},\Z_q)\le \max\{35q^2,qk+q-1\}$, leaving an additive gap of order $q/2$ when $k$ is large. Moreover, for the case $q=3$, we prove that \(R(C_{3k}, \mathbb{Z}_3) = 3k + 2\) for all \(k \ge 2\). Extending our techniques beyond cycles, we also resolve the zero-sum Ramsey number for wheel graphs \(W_m = C_m + K_1\), proving that \(R(W_{3k}, \mathbb{Z}_3) = 3k + 1\) for all \(k \ge 2\).