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arXiv · 2505.24577

The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture

Francesco Barioli·Shaun M. Fallat·Himanshu Gupta·Zhongshan Li
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Abstract

Since the transformative workshop by the American Institute of Mathematics on the minimum rank of a graph, two longstanding open problems have captivated the community interested in the minimum rank of graphs: the graph complement conjecture and the $δ$-conjecture. In this paper, we use a classical result of Mader (1972) to establish a weak version of the graph complement conjecture for all key minimum rank parameters. In addition, again using the same result of Mader, we present some extremal resolutions of the $δ$-conjecture. Furthermore, we incorporate the assumption of the $δ$-conjecture and extensive work on graph degeneracy to improve the bound in the weak version of the graph complement conjecture. We conclude with a list of conjectured bounds on the positive semidefinite variant of the Colin de Verdière number.

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Francesco Barioli, Shaun M. Fallat, Himanshu Gupta, Zhongshan Li. 2025-05-30. The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture. https://arxiv.org/abs/2505.24577

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