  A graph G is an NG-graph if \chi(G) + \chi(G complement) = |V(G)| + 1. We
characterize NG-graphs solely from degree sequences leading to a linear-time
recognition algorithm. We also explore the connections between NG-graphs and
split graphs. There are three types of NG-graphs and split graphs can also be
divided naturally into two categories, balanced and unbalanced. We characterize
each of these five classes by degree sequence. We construct bijections between
classes of NG-graphs and balanced and unbalanced split graphs which, together
with the known formula for the number of split graphs on n vertices, allows us
to compute the sizes of each of these classes. Finally, we provide a bijection
between unbalanced split graphs on n vertices and split graphs on n-1 or fewer
vertices providing evidence for our conjecture that the rapid growth in the
number of split graphs comes from the balanced split graphs.
