Nathanson's Heights and the CSS Conjecture for Cayley Graphs

TL;DR

This paper extends the proof of the CSS conjecture for Cayley graphs of cyclic groups from prime to all positive integers by generalizing the concept of height.

Contribution

It generalizes the definition of height and proves the CSS conjecture for Cayley graphs of cyclic groups for any positive integer N.

Findings

CSS conjecture holds for Cayley graphs of cyclic groups for all positive integers N.

Extension of height concept to composite N.

Verification of the conjecture beyond prime N.

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 $N$.