An extremal problem on graphic sequences with a realization containing every ℓ-tree on k vertices

De-Yan Zeng and Jian-Hua Yin

Applied Mathematics and Computation, 2018, vol. 337, issue C, 487-493

Abstract: A graph G is a ℓ-tree if G=Kℓ+1, or G has a vertex v whose neighborhood is a clique of order ℓ, and G−v is a ℓ-tree. A non-increasing sequence π=(d1,…,dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. Yin and Li (2009) proved that if k ≥ 2, n≥92k2+192k and π=(d1,…,dn) is a graphic sequence with ∑i=1ndi>(k−2)n, then π has a realization containing every 1-tree (the usual tree) on k vertices. Moreover, the lower bound (k−2)n is the best possible. This is a variation of a conjecture due to Erdős and Sós. Recently, Zeng and Yin (2016) investigated an analogue extremal problem for 2-trees and prove that if k ≥ 3, n≥2k2−k and π=(d1,…,dn) is a graphic sequence with ∑i=1ndi>4k−53n, then π has a realization containing every 2-tree on k vertices. Moreover, the lower bound 4k−53n is almost the best possible. In this paper, we consider the most general case ℓ ≥ 3 and prove that if ℓ ≥ 3, k≥ℓ+1,n≥2k2−ℓk+k and π=(d1,…,dn) is a graphic sequence with ∑i=1ndi>2ℓk−ℓ−3ℓ+1n, then π has a realization containing every ℓ-tree on k vertices. We also show that the lower bound 2ℓk−ℓ−3ℓ+1n is almost the best possible.

Keywords: Degree sequences; Graphic sequences; Realization; ℓ-trees (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300318304727
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:337:y:2018:i:c:p:487-493

DOI: 10.1016/j.amc.2018.05.055

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().