Solution to a general version of a degree sequence variant of the Erdős–Sós conjecture

Abstract

Let P denote a graph property, such as “being k-connected”, “being hamiltonian”, or “containing H as a subgraph”, etc. A graphic sequence π is potentially P-graphic if there is a realization of π having the property P. In 1991, Erdős et al. introduced the following problem: Determine the minimum even integer σ(P,n) such that every n-term graphic sequence with sum at least σ(P,n) is potentially P-graphic. The parameter σ(P,n) is known as the potential function of P, and can be viewed as a degree sequence variant of the classical extremal function ex(P,n). A simple graph G is an ℓ-tree if G=Kℓ+1, or G has a vertex whose neighborhood is a clique of order ℓ, and G−v is an ℓ-tree. Clearly, 1-trees are the usual trees. For ℓ≥1 and k≥ℓ+1, let P(ℓ,k) be the graph property “containing every ℓ-tree on k vertices as a subgraph”. For ℓ=1, Yin and Li determined σ(P(1,k),n) for k≥2 and n≥92k2+192k. This is a degree sequence variant of the Erdős–Sós conjecture. For ℓ=2, Zeng et al. determined σ(P(2,k),n) for k≥3 and n sufficiently large. In this paper, we consider the most general case ℓ≥3, and determine σ(P(ℓ,k),n) for ℓ≥3, k≥ℓ+1 and n sufficiently large.