Packing tools

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A recently located independence bound

Abiad, Kumar, and Pragada, Localization of the Caro--Wei bound and its applications to bipartiteness, arXiv:2609.00210v1 (31 August 2026), Theorems 2.2 and 3.1, prove

$$\alpha(H)\ge\sum_{v\in V(H)}\frac{2}{d_H(v)+c_H(v)+1},$$

where $c_H(v)$ is the largest clique order through $v$. The inequality portion of their quadratic minimization/private-neighbor proof was read and checked by the author. The equality classification is not needed here. Jensen's inequality gives

$$e(H)\ge\frac{|H|^2}{\alpha(H)} -\frac{(\omega(H)+1)|H|}{2}. \tag{1}$$

This input is outside the originally supplied source bundle; its provenance is therefore explicit rather than silently treated as an old known theorem.

In averaged form, Jensen and $c(v)\le\omega(H)$ give

$$\alpha(H)\ge\frac{2|H|}{\overline d(H)+\omega(H)+1}.$$

The primary paper's Theorem 2.2 inequality proof and Theorem 3.1 statement were rechecked.