A linear upper bound on zero-sum Ramsey numbers of $d$-degenerate graphs in ${\mathbb{Z}}_p$

Abstract

Let $p$ be a prime number and let $G$ be a graph on $n$ vertices and $m$ edges. The zero-sum Ramsey number of $G$ over ${\mathbb{Z}}_p$, denoted by $R(G,{\mathbb{Z}}_p)$, is the minimum $\ell \in \mathbb{N}$ such that for any edge-coloring $c:E(K_{\ell })\to {\mathbb{Z}}_p$, there is a subgraph $G^{\prime }\subset K_{\ell }$ isomorphic to $G$ and satisfying ${\sum }_{e\in E(G^{\prime })}c(e)=0$. We prove that if $G$ is a $d$-degenerate graph, then $R(G,{\mathbb{Z}}_p)\le n+(3+3d)p$ so long as $m\ge 2pd(d+1{)}^2$, $p$ divides $m$, and $2d<p$. This generalizes a result by Colucci and D'Emidio on $1$-degenerate graphs.