Nathanson Heights and the CSS Conjecture for Cayley Graphs
Yotsanan Meemark, Chaiwat Pinthubthaworn   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 / 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 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\gamma(G)/2. Nathanson and Sullivan verified
this conjecture for the directed Cayley graph Cay(Z/NZ, EA)
whose vertex set is the additive group Z/NZ and whose edge set
EA is determined by EA = {(x, x +a) : x \in Z/NZ, a \in A} when
N is prime and |A| \leq (N -1)/4 by introducing “height”. In
this work, we extend the definition of height and apply to
answer the CSS conjecture for Cay(Z/NZ, EA) to any positive
integer N and |A|\leq (N-1)/4.  Full