On induced subgraph of Cartesian product of paths

Abstract

Chung et al. constructed an induced subgraph of the hypercube $Q^n$ with $\alpha \left(Q^n\right)+1$ vertices and with maximum degree smaller than $\lceil \sqrt{n}\rceil$. Subsequently, Huang 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 $f\left(G\right)$ be the minimum of the maximum degree of an induced subgraph of $G$ on $\alpha \left(G\right)+1$ vertices, what can we say about $f\left(G\right)$? 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 $f\left(P_m^k\right)$ when $m=2n+1$ by showing that $f\left(P_{2n+1}^k\right)=1$ for $n\ge 2$ and $f\left(P_3^k\right)=2$, and give a nontrivial lower bound of $f\left(P_m^k\right)$ when $m=2n$ by showing that $f\left(P_{2n}^k\right)\ge \lceil 2\cos \frac{\pi n}{2n+1}\sqrt{k}\rceil$. In particular, when $n=1$, we have $f\left(Q^k\right)=f\left(P_2^k\right)\ge \sqrt{k}$, which is Huang's result. The lower bounds of $f\left(P_3^k\right)$ and $f\left(P_{2n}^k\right)$ are given by using the spectral method provided by Huang.