Counting independent sets in regular graphs with bounded independence number
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910670580088832 |
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| author | Galvin, David Marmorino, Phillip |
| author_facet | Galvin, David Marmorino, Phillip |
| contents | An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $α$.
We give upper and lower bounds that in many cases are close to each other. In particular, for each $0 < c_{\rm ind} \leq 1/2$ we exhibit a constant $k(c_{\rm ind})$ such that if $(G_n)_{n \in {\mathbb N}}$ is a sequence of graphs with $G_n$ $d$-regular on $n$ vertices and with maximum independent set size at most $α$, with $d\rightarrow \infty$ and $α/n \rightarrow c_{\rm ind}$ as $n \rightarrow \infty$, then $G_n$ has at most $k(c_{\rm ind})^{n+o(n)}$ independent sets, and we show that there is a sequence $(G_n)_{n \in {\mathbb N}}$ of graphs with $G_n$ $d$-regular on $n$ vertices ($d \leq n/2$) and with maximum independent set size at most $α$, with $α/n \rightarrow c_{\rm ind}$ as $n \rightarrow \infty$ and with $G_n$ having at least $k(c_{\rm ind})^{n+o(n)}$ independent sets. We also consider the regime $1/2 < c_{\rm ind} < 1$. Here for each $0 < c_{\rm deg} \leq 1-c_{\rm ind}$ we exhibit a constant $k(c_{\rm ind},c_{\rm deg})$ for which an analogous pair of statements can be proven, except that in each case we add the condition $d/n \rightarrow c_{\rm deg}$ as $n \rightarrow \infty$.
Our upper bounds are based on graph container arguments, while our lower bounds are constructive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19959 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting independent sets in regular graphs with bounded independence number Galvin, David Marmorino, Phillip Combinatorics 05C69 An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $α$. We give upper and lower bounds that in many cases are close to each other. In particular, for each $0 < c_{\rm ind} \leq 1/2$ we exhibit a constant $k(c_{\rm ind})$ such that if $(G_n)_{n \in {\mathbb N}}$ is a sequence of graphs with $G_n$ $d$-regular on $n$ vertices and with maximum independent set size at most $α$, with $d\rightarrow \infty$ and $α/n \rightarrow c_{\rm ind}$ as $n \rightarrow \infty$, then $G_n$ has at most $k(c_{\rm ind})^{n+o(n)}$ independent sets, and we show that there is a sequence $(G_n)_{n \in {\mathbb N}}$ of graphs with $G_n$ $d$-regular on $n$ vertices ($d \leq n/2$) and with maximum independent set size at most $α$, with $α/n \rightarrow c_{\rm ind}$ as $n \rightarrow \infty$ and with $G_n$ having at least $k(c_{\rm ind})^{n+o(n)}$ independent sets. We also consider the regime $1/2 < c_{\rm ind} < 1$. Here for each $0 < c_{\rm deg} \leq 1-c_{\rm ind}$ we exhibit a constant $k(c_{\rm ind},c_{\rm deg})$ for which an analogous pair of statements can be proven, except that in each case we add the condition $d/n \rightarrow c_{\rm deg}$ as $n \rightarrow \infty$. Our upper bounds are based on graph container arguments, while our lower bounds are constructive. |
| title | Counting independent sets in regular graphs with bounded independence number |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2410.19959 |