Ratio bound (Lov\'asz number) versus inertia bound

Ferdinand Ihringer

Ratio bound (Lov\'asz number) versus inertia bound

Combinatorics 2025-05-14 v2

Authors: Ferdinand Ihringer

Abstract

Matthew Kwan and Yuval Wigderson showed that for an infinite family of graphs, the Lov\'asz number gives an upper bound of $O(n^{3/4})$ for the size of an independent set (where $n$ is the number of vertices), while the weighted inertia bound cannot do better than $\Omega(n)$ . Here we point out that there is an infinite family of graphs for which the Lov\'asz number is $\Omega(n^{3/4})$ , while the unweighted inertia bound is $O(n^{1/2})$ .

Keywords

Lovász number inertia bound independent set graph theory ratio bound

Cite

@article{arxiv.2312.09524,
  title  = {Ratio bound (Lov\'asz number) versus inertia bound},
  author = {Ferdinand Ihringer},
  journal= {arXiv preprint arXiv:2312.09524},
  year   = {2025}
}

Comments

4 pages. I do not plan to publish this note/example

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