Nathanson's Heights and the CSS Conjecture for Cayley Graphs
| dc.creator | Meemark, Yotsanan | |
| dc.creator | Pinthubthaworn, Chaiwat | |
| dc.date | 2008-05-03 | |
| dc.date.accessioned | 2026-07-07T09:36:52Z | |
| dc.date.available | 2026-07-07T09:36:52Z | |
| dc.description | Let $G$ be a finite directed graph, $β(G)$ the minimum size of a subset $X$ of edges such that the graph $G' = (V,E \smallsetminus X)$ is directed acyclic and $γ(G)$ 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 $β(G) \leq γ(G)$. They conjectured a sharper bound (so called the "CSS conjecture") that $β(G) \leq \dfrac{γ(G)}{2}$. 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 $N$. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/0805.0341 | |
| dc.identifier | http://arxiv.org/abs/0805.0341 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160272 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 05C25, 11A07 | |
| dc.title | Nathanson's Heights and the CSS Conjecture for Cayley Graphs | |
| dc.type | text |