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arXiv · 2609.06312

On some k-fold generalizations of Lov\'asz theta and their sandwich theorems

Marcel K Carli Silva·Gabriel Coutinho·Thiago Oliveira·Levent Tunçel
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Abstract

We study several $k$-fold generalizations of the Lov\'asz theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan--Manber parameter $\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $\varphi_k$ obtained from orthonormal representations of graphs and show the inequality $\varphi_k \leq \vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\vartheta_k$, $\varphi_k$, $\vartheta(G\square K_k)$, and related convexifications.

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Marcel K Carli Silva, Gabriel Coutinho, Thiago Oliveira, Levent Tunçel. 2026-09-06. On some k-fold generalizations of Lov\'asz theta and their sandwich theorems. https://arxiv.org/abs/2609.06312

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