On induced subgraph of Cartesian product of paths

Chung et al. constructed an induced subgraph of the hypercube Qn with a (Qn) + 1 vertices and with maximum degree smaller than. n.. Subsequently, Huang proved the Sensitivity Conjecture by demonstrating that the maximum degree of such an induced subgraph of hypercube Qn is at least. n., 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 a G()+ 1 vertices, what can we say about f G()? In this paper, we investigate this question for Cartesian product of paths Pm, denoted by Pmk. We determine the exact values of f P() m k when m = 2n + 1 by showing that f (P n) = 1k 2 +1 for n = 2 and f (P) = 2k 3, and give a nontrivial lower bound of f P() m k when m = 2n by showing that f (P n) =.2 cos k.k pn 2 2n + 1. In particular, when n = 1, we have f (Qk) = f (Pk) = k 2, which is Huang's result. The lower bounds of f (Pk) 3 and f (P n)k 2 are given by using the spectral method provided by Huang.