On Induced Subgraph of Cartesian Product of Paths

Chung, F\"uredi, Graham, and Seymour (JCTA, 1988) constructed an induced subgraph of the hypercube $Q^n$ with $\alpha(Q^n)+1$ vertices and with maximum degree smaller than $\lceil \sqrt{n} \rceil$. Subsequently, Huang (Annals of Mathematics, 2019) proved the Sensitivity Conjecture by demonstrating that the maximum degree of such an induced subgraph of hypercube $Q^n$ is at least $\lceil \sqrt{n} \rceil$, and posed the question: Given a graph $G$, let $f(G)$ be the minimum of the maximum degree of an induced subgraph of $G$ on $\alpha(G)+1$ vertices, what can we say about $f(G)$? In this paper, we investigate this question for Cartesian product of paths $P_m$, denoted by $P_m^k$. We determine the exact values of $f(P_{m}^k)$ when $m=2n+1$ by showing that $f(P_{2n+1}^k)=1$ for $n\geq 2$ and $f(P_3^k)=2$, and give a nontrivial lower bound of $f(P_{m}^k)$ when $m=2n$ by showing that $f(P_{2n}^k)\geq \lceil \sqrt{\beta_nk}\rceil$. In particular, when $n=1$, we have $f(Q^k)=f(P_{2}^k)\ge \sqrt{k}$, which is Huang's result. The lower bounds of $f(P_{3}^k)$ and $f(P_{2n}^k)$ are given by using the spectral method provided by Huang.