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

• Published in: Discrete mathematics Vol. 312; no. 9; pp. 1494 - 1501
• Main Authors: Barrus, Michael D., Hartke, Stephen G., Jao, Kyle F., West, Douglas B.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 06.05.2012
• Subjects: Aigner–Triesch method; Constrictions; Dominance order; Erdős–Gallai inequalities; Gaps; Graphic list; Graphic sequence; Inequalities; Integer partition; Integers; Lists; Mathematical analysis; Thresholds; Graphic list; Graphic sequence; Aigner–Triesch method; Dominance order; Erdős–Gallai inequalities; Integer partition
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2011.05.001

In a list (d1,…,dn) 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 ℓth Erdős–Gallai inequality, where ℓ=max{k:dk≥k}.