Affiliation:
1. School of the Gifted Young University of Science and Technology of China Hefei Anhui China
2. School of Mathematical Sciences Shanghai Jiao Tong University Shanghai China
3. School of Mathematical Sciences University of Science and Technology of China Hefei Anhui China
4. CAS Key Laboratory of Wu Wen‐Tsun Mathematics University of Science and Technology of China Hefei Anhui China
5. Hefei National Laboratory University of Science and Technology of China Hefei Anhui China
Abstract
AbstractChung et al. constructed an induced subgraph of the hypercube with vertices and with maximum degree smaller than . Subsequently, Huang proved the Sensitivity Conjecture by demonstrating that the maximum degree of such an induced subgraph of hypercube is at least , and posed the question: Given a graph , let be the minimum of the maximum degree of an induced subgraph of on vertices, what can we say about ? In this paper, we investigate this question for Cartesian product of paths , denoted by . We determine the exact values of when by showing that for and , and give a nontrivial lower bound of when by showing that . In particular, when , we have , which is Huang's result. The lower bounds of and are given by using the spectral method provided by Huang.
Reference14 articles.
1. Unitary signings and induced subgraphs of Cayley graphs of Zn ${Z}_{n}$;Alon N.;Adv. Combin,2020
2. On induced subgraphs of the cube
3. On induced subgraphs of the Hamming graph
4. A very short proof of Cauchys interlace theorem for eigenvalues of Hermitian matrices;Fisk S.;Amer. Math. Monthly,2005
5. On products and line graphs of signed graphs, their eigenvalues and energy