  For an integer sequence (with even sum), the closer that the sequence is to
being regular, the more likely that the sequence is graphic. But how regular
must a sequence be before it must always be graphic? We show that for many
sequences if all values are within $\frac{n-2}{4}$ of the mean degree value,
then the sequence is graphic. We also see how this result extends to show when
a maximum difference between sequence values implies that a sequence is
graphic.
