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.

中文翻译：

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可分割细分的严格界限

Alon 和 Krivelevich 证明了对于每个n顶点次立方图H和每个整数$q\ge 2$存在一个（最小的）整数$F=F（H,q）$使得每个$K_F$-minor 包含H的细分，其中每个细分路径的长度可被q整除。改善他们的超指数界限，我们证明$F（H,q）\le \frac{21}{2}qn+8n+14q$，在乘法因子为常数的范围内是最佳的。