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arXiv · 2609.00210

Localization of the Caro-Wei bound and its applications to bipartiteness

Aida Abiad·Hitesh Kumar·Shivaramakrishna Pragada
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Abstract

We confirm a conjecture of Brause, Randerath, Rautenbach and Schiermeyer (2016) by proving a localized lower bound on the independence number of a graph that strengthens the classical bounds of Fajtlowicz (1978) and of Caro (1979) and Wei (1981), which in turn settles a conjecture by Bertram and Hor\'{a}k (1996). Our proof is based on a new Motzkin--Straus-type inequality involving local clique numbers and the independence number. We then apply the developed methods to study spectral and algebraic measures of graph bipartiteness. In particular, we extend a theorem of Brandt (1998) on spectral bipartiteness from regular $K_{r+1}$-free graphs to all $K_{r+1}$-free graphs, we improve a general upper bound for the least signless Laplacian eigenvalue of $K_{r+1}$-free graphs, and we disprove a conjecture of de Lima, Nikiforov and Oliveira (2016) in the case of $K_4$-free graphs.

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Aida Abiad, Hitesh Kumar, Shivaramakrishna Pragada. 2026-08-31. Localization of the Caro-Wei bound and its applications to bipartiteness. https://arxiv.org/abs/2609.00210

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