On zero-sum Ramsey numbers of cycles and wheels

Abstract

For an integer $q\ge 2$ and a graph $F$ with $q\mid e(F)$, let $R(F,{\mathbb{Z}}_q)$ be the least integer $n$ such that every edge-labeling $w:E(K_n)\to {\mathbb{Z}}_q$ contains a copy of $F$ whose edge-label sum is zero in ${\mathbb{Z}}_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},{\mathbb{Z}}_q)\le \max \{R(C_{2q},{\mathbb{Z}}_q),qk+q-1\}$ via an insertion argument rooted in the classic Erdős-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},{\mathbb{Z}}_q)\le \max \{35q^2,qk+q-1\}$ for every $q\ge 3$. We also show that $R(C_{qk},{\mathbb{Z}}_q)\ge qk+q-1$ for odd $q\ge 3$. Hence, for every fixed odd $q\ge 3$ and every $k\ge 35q$, we obtain the exact value $R(C_{qk},{\mathbb{Z}}_q)=qk+q-1$. For even $q\ge 4$, the same method gives $qk+\frac{q}{2}-1\le R(C_{qk},{\mathbb{Z}}_q)\le \max \{35q^2,qk+q-1\}$, 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 \(k \ge 2\).