Tight bounds for divisible subdivisions

Abstract

Alon and Krivelevich proved that for every n-vertex subcubic graph H and every integer $q\ge 2$ there exists a (smallest) integer $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of H in which the length of every subdivision-path is divisible by q. Improving their superexponential bound, we show that $f(H,q)\le \frac{21}{2}qn+8n+14q$, which is optimal up to a constant multiplicative factor.