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

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.