A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs

Let $G$ be a graph and $\mathrm{\Gamma }$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\mathrm{\Gamma }$, denoted by $R(G,\mathrm{\Gamma })$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to \mathrm{\Gamma }$ contains a copy of $G$ with ${\sum }_{e\in E(G)}c(e)=0_{\mathrm{\Gamma }}$. We prove a linear upper bound $R(G,\mathrm{\Gamma })\le Cn$ that holds for every $n$-vertex graph $G$ with bounded maximum degree and every finite abelian group $\mathrm{\Gamma }$ with $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }$ dividing $e(G)$.