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 $G'\subset K_\ell$ isomorphic to $G$ and satisfying $\sum_{e\in E(G')}c(e)=0$. We prove that if $G$ is a $d$-degenerate graph, then $R(G, \mathbb{Z}_p)\leq n + (3+3d)p$ so long as $m\geq 2pd(d+1)^2$, $p$ divides $m$, and $2d<p$. This generalizes a result by Colucci and D'Emidio on $1$-degenerate graphs.