## Abstract

Given a graph *H*, a graphic sequence π is *potentially H-graphic* if there is some realization of π that contains *H* as a subgraph. In 1991, Erdős, Jacobson and Lehel posed the following question:

Determine the minimum even integer σ(*H,n*) such that every *n*-term graphic sequence with sum at least σ(*H,n*) is potentially *H*-graphic. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problems.

While the exact value of σ(*H,n*) has been determined for a number of specific classes of graphs (including cliques, cycles, complete bigraphs and others), very little is known about the parameter for arbitrary *H*. In this paper, we determine σ(*H,n*) asymptotically for all *H*, thereby providing an Erdős-Stone-Simonovits-type theorem for the Erdős-Jacobson-Lehel problem.

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Research Supported in part by Simons Foundation Collaboration Grant #206692.

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Ferrara, M.J., Lesaulnier, T.D., Moffatt, C.K. *et al.* On the sum necessary to ensure that a degree sequence is potentially *H*-graphic.
*Combinatorica* **36, **687–702 (2016). https://doi.org/10.1007/s00493-015-2986-1

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### Mathematics Subject Classification (2000)

- 05C07
- 05C35