Minimum degree stability of H-free graphs
<p>Given an (r + 1)-chromatic graph H, the fundamental edge stability result of Erdős and Simonovits says that all n-vertex H-free graphs have at most (1 − 1/r + o(1))( n 2 ) edges, and any H-free graph with that many edges can be made r-partite by deleting o(n 2 ) edges. Here we con...
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| Format: | Journal article |
| Language: | English |
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Springer
2023
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| Summary: | <p>Given an (r + 1)-chromatic graph H, the fundamental edge stability result of Erdős and Simonovits says that all n-vertex H-free graphs have at most (1 − 1/r + o(1))( n 2 ) edges, and any H-free graph with that many edges can be made r-partite by deleting o(n 2 ) edges. Here we consider a natural variant of this – the minimum degree stability of H-free graphs. In particular, what is the least c such that any n-vertex H-free graph with minimum degree greater than cn can be made r-partite by deleting o(n 2 ) edges? We determine this least value for all 3- chromatic H and for very many non-3-colourable H (all those in which one is commonly interested) as well as bounding it for the remainder. This extends the Andrásfai-Erdős-Sós theorem and work of Alon and Sudakov</p> |
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