On extremal numbers of the triangle plus the four-cycle
For a family $\mathcal {F}$ of graphs, let ${\mathrm {ex}}(n,\mathcal {F})$ denote the maximum number of edges in an n-vertex graph which contains none of the members of $\mathcal {F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function...
| Published in: | Forum of Mathematics, Sigma |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942510100X/type/journal_article |
| Summary: | For a family
$\mathcal {F}$
of graphs, let
${\mathrm {ex}}(n,\mathcal {F})$
denote the maximum number of edges in an n-vertex graph which contains none of the members of
$\mathcal {F}$
as a subgraph. A longstanding problem in extremal graph theory asks to determine the function
${\mathrm {ex}}(n,\{C_3,C_4\})$
. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of
${\mathrm {ex}}(n,\{C_3,C_4\})$
since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3]. |
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| ISSN: | 2050-5094 |