  There are a variety of existing conditions for a degree sequence to be
graphic. When a degree sequence satisfies any of these conditions, there exists
a graph that realizes the sequence. We formulate several novel sufficient
graphicality criteria that depend on the number of elements in the sequence,
corresponding to the number of nodes in an associated graph, and the mean
degree of the sequence. These conditions, which are stated in terms of bidegree
sequences for directed graphs, are easier to apply than classic necessary and
sufficient graphicality conditions involving multiple inequalities. They are
also more flexible than more recent graphicality conditions, in that they imply
graphicality of some degree sequences not covered by those conditions. The form
of our results will allow them to be easily used for the generation of graphs
with particular degree sequences for applications.
