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

In a list (d1,…,dn)(d1,…,dn) of positive integers, let rr and ss denote the largest and smallest entries. A list is gap-free   if each integer between rr and ss 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 gg, we prove a more general length threshold that includes both of these results. As a tool, we prove that if a positive list dd with even sum has no repeated entries other than rr and ss (and the length exceeds rr), then to prove that dd is graphic it suffices to check only the ℓℓth Erdős–Gallai inequality, where ℓ=max{k:dk≥k}.