We discuss supergraphs and their relation to "chiral functions" in N=4 Super
Yang-Mills. Based on the magnon dispersion relation and an explicit three-loop
result of Sieg's we make an all loop conjecture for the rational contributions
of certain classes of supergraphs. We then apply superspace techniques to the
"cubic" branch of Leigh-Strassler N=1 superconformal theories. We show that
there are order 2^L/L single trace operators of length L which have zero
anomalous dimensions to all loop order in the planar limit. We then compute the
anomalous dimensions for another class of single trace operators we call
one-pair states. Using the conjecture we can find a simple expression for the
rational part of the anomalous dimension which we argue is valid at least up to
and including five-loop order. Based on an explicit computation we can compute
the anomalous dimension for these operators to four loops.Comment: 22 pages; v2: Conjecture modified to apply only for the rational part
of the chiral functions. Typos fixed. Minor modification