Nathanson's Heights and the CSS Conjecture for Cayley Graphs

Abstract

Let $G$ be a finite directed graph, $\beta(G)$ the minimum size of a subset $X$ of edges such that the graph $G' = (V,E \smallsetminus X)$ 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 $\beta(G) \leq \gamma(G)$. 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 $\Cay(\bbZ/N\bbZ, E_A)$ whose vertex set is the additive group $\bbZ/N\bbZ$ 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 $\Cay(\bbZ/N\bbZ, E_A)$ to any positive integer $N$.