Length thresholds for graphic lists given fixed largest and smallest entries and bounded gaps

In a list (d"1,...,d"n) of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. We prove that a gap-free list with even sum is graphic if it has at least r+r+s+12s terms. With no restriction on gaps, length at least (r+s+1)^24s suffices, as proved by Zverovich and Zverovich (1992). Both bounds are sharp within 1. When the gaps between consecutive terms are bounded by g, we prove a more general length threshold that includes both of these results. As a tool, we prove that if a positive list d with even sum has no repeated entries other than r and s (and the length exceeds r), then to prove that d is graphic it suffices to check only the @?th Erdos-Gallai inequality, where @?=max{k:d"k>=k}.