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On the sum necessary to ensure that a degree sequence is potentially H-graphic

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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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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO, 80217, USA

    Michael J. Ferrara & Casey K. Moffatt

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Timothy D. Lesaulnier

  3. School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Paul S. Wenger

Authors
  1. Michael J. Ferrara
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  2. Timothy D. Lesaulnier
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  3. Casey K. Moffatt
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  4. Paul S. Wenger
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Corresponding author

Correspondence to Michael J. Ferrara.

Additional information

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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  • DOI: https://doi.org/10.1007/s00493-015-2986-1

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