Sparse Sets in Triangle-free Graphs
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arXiv
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| Format: | Preprint |
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2024
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| author | Ekim, Tınaz Erdem, Burak Nur Gimbel, John |
| author_facet | Ekim, Tınaz Erdem, Burak Nur Gimbel, John |
| contents | A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every triangle-free graph of order 11 contains a 1-sparse 5-set; every triangle-free graph of order 13 contains a 2-sparse 7-set; and every triangle-free graph of order 8 contains a 3-sparse 6-set. Further, these are all best possible.
For fixed $k$, we consider the growth rate of the largest $k$-sparse set of a triangle-free graph of order $n$. Also, we consider Ramsey numbers of the following type. Given $i$, what is the smallest $n$ having the property that all triangle-free graphs of order $n$ contain a 4-cycle or a $k$-sparse set of order $i$. We use both direct proof techniques and an efficient graph enumeration algorithm to obtain several values for defective Ramsey numbers and a parameter related to largest sparse sets in triangle-free graphs, along with their extremal graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparse Sets in Triangle-free Graphs Ekim, Tınaz Erdem, Burak Nur Gimbel, John Combinatorics Discrete Mathematics 05C30, 05C35, 05C55 A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every triangle-free graph of order 11 contains a 1-sparse 5-set; every triangle-free graph of order 13 contains a 2-sparse 7-set; and every triangle-free graph of order 8 contains a 3-sparse 6-set. Further, these are all best possible. For fixed $k$, we consider the growth rate of the largest $k$-sparse set of a triangle-free graph of order $n$. Also, we consider Ramsey numbers of the following type. Given $i$, what is the smallest $n$ having the property that all triangle-free graphs of order $n$ contain a 4-cycle or a $k$-sparse set of order $i$. We use both direct proof techniques and an efficient graph enumeration algorithm to obtain several values for defective Ramsey numbers and a parameter related to largest sparse sets in triangle-free graphs, along with their extremal graphs. |
| title | Sparse Sets in Triangle-free Graphs |
| topic | Combinatorics Discrete Mathematics 05C30, 05C35, 05C55 |
| url | https://arxiv.org/abs/2406.03290 |