A linear upper bound on the ${\mathbb{Z}}_p$-Ramsey number of graphs with sufficiently large $2$-packing

Given a positive integer $k$ and graph $G$, the ${\mathbb{Z}}_k$-Ramsey number $R(G,{\mathbb{Z}}_k)$ is the least $N$ (if it exists) such that every coloring $f:E(K_N)\to {\mathbb{Z}}_k$ contains a copy $G^{\mathrm{\prime }}$ of $G$ such that ${\sum }_{e\in E(G^{\mathrm{\prime }})}f(e)=0$. Motivated by a question of Caro and Mifsud, we study the ${\mathbb{Z}}_k$-Ramsey number of graphs with a sufficiently large 2-packing, i.e. a set of vertices $S\subseteq V(G)$ such that $N[u]\cap N[v]=\mathrm{\varnothing }$ for all distinct $u,v\in S$. In particular, we prove that $R(G,{\mathbb{Z}}_p)\le n+6p-9$ for all $n$-vertex graphs $G$ and all primes $p$ such that $p$ divides $e(G)$, the minimum degree of $G$ is at least $1$, and there exists a $2$-packing of $G$ with size $p-1$. This upper bound improves depending on vertex degrees in the $2$-packing, with equality in certain cases. The result also implies an upper bound of the form $R(G,{\mathbb{Z}}_p)\le n+C$ for $n$-vertex graphs $G$ of bounded maximum degree.