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 + 1 2 s terms. With no restriction on gaps, length at least ( r + s + 1 ) 2 4 s 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 ℓ th Erdős–Gallai inequality, where ℓ = max { k : d k ≥ k } .