Abstract
Let \(h=\prod _{i=1}^{t}p_i^{s_i}\) be its decomposition into a product of powers of distinct primes, and \(\mathbb {Z}_{h}\) be the residue class ring modulo h. Let \(1\le r\le \min \{m,n\}\) and \(\mathbb {Z}_{h}^{m\times n}\) be the set of all \(m\times n\) matrices over \(\mathbb {Z}_{h}\). The generalized bilinear forms graph over \(\mathbb {Z}_{h}\), denoted by \(\hbox {Bil}_r\left( \mathbb {Z}_{h}^{m\times n}\right) \), has the vertex set \(\mathbb {Z}_{h}^{m\times n}\), and two distinct vertices A and B are adjacent if the inner rank of \(A-B\) is less than or equal to r. In this paper, we determine the clique number and geometric structures of maximum cliques of \(\hbox {Bil}_r\left( \mathbb {Z}_{h}^{m\times n}\right) \). As a result, the Erdős-Ko-Rado theorem for \(\mathbb {Z}_h^{m\times n}\) is obtained.
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The author is indebted to the anonymous reviewers for their detailed reports and constructive suggestions.
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This research is supported by National Natural Science Foundation of China (11971146).
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Guo, J. Erdős-Ko-Rado Theorem for Matrices Over Residue Class Rings. Graphs and Combinatorics 37, 2497–2510 (2021). https://doi.org/10.1007/s00373-021-02371-0
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DOI: https://doi.org/10.1007/s00373-021-02371-0
Keywords
- Erdős-Ko-Rado theorem
- Residue class ring
- Bilinear forms graph
- Clique number
- Independence number
- Chromatic number