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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2606.01761 |
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| _version_ | 1866910279902691328 |
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| author | Gao, Jun Ma, Jie Pikhurko, Oleg |
| author_facet | Gao, Jun Ma, Jie Pikhurko, Oleg |
| contents | The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\ge 2$.
Continuing a line of work comparing these two parameters with other known bounds, we show that the $\vartheta$-function of L. Lovász and the weighted inertia bound of A. R. Calderbank and P. Frankl, when applied directly to $G^k$, perform at least as well as the ratio and inertia bounds of Abiad-Coutinho-Fiol, respectively. In particular, $\vartheta(G^k)$ provides a polynomial-time computable upper bound on the independence number of $G^k$ that is at least as strong as the ratio bound when the latter applies (i.e.,\ when the graph $G$ is regular). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01761 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on the Ratio and Inertia Bounds for the $k$-Independence Number Gao, Jun Ma, Jie Pikhurko, Oleg Combinatorics The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\ge 2$. Continuing a line of work comparing these two parameters with other known bounds, we show that the $\vartheta$-function of L. Lovász and the weighted inertia bound of A. R. Calderbank and P. Frankl, when applied directly to $G^k$, perform at least as well as the ratio and inertia bounds of Abiad-Coutinho-Fiol, respectively. In particular, $\vartheta(G^k)$ provides a polynomial-time computable upper bound on the independence number of $G^k$ that is at least as strong as the ratio bound when the latter applies (i.e.,\ when the graph $G$ is regular). |
| title | A note on the Ratio and Inertia Bounds for the $k$-Independence Number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.01761 |