Nathanson's Heights and the CSS Conjecture for Cayley Graphs

Let $G$ be a finite directed graph, $\beta (G)$ the minimum size of a subset $X$ of edges such that the graph $G^{\mathrm{\prime }}=(V,E\setminus X)$ is directed acyclic and $\gamma (G)$ the number of pairs of nonadjacent vertices in the undirected graph obtained from $G$ by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan \cite{CSS07} proved that if $G$ is triangle-free, then $\beta (G)\le \gamma (G)$. They conjectured a sharper bound (so called the “CSS conjecture”) that $\beta(G) \leq \dfrac{\gamma(G)}{2}$. Nathanson and Sullivan verified this conjecture for the directed Cayley graph $\text{Cay}(\text{bbZ}\mathrm{/}N\text{bbZ},E_A)$ whose vertex set is the additive group $\text{bbZ}\mathrm{/}N\text{bbZ}$ and whose edge set $E_A$ is determined by $E_A = {(x,x+a) : x \in \bbZ/N\bbZ, a \in A}$when$N$ is prime in \cite{NS07} by introducing “height”. In this work, we extend the definition of height and the proof of CSS conjecture for $\text{Cay}(\text{bbZ}\mathrm{/}N\text{bbZ},E_A)$ to any positive integer $N$.