Tight bounds for divisible subdivisions

TL;DR

This paper improves bounds on the size of complete graphs needed to contain subdivisions of certain graphs with path lengths divisible by a given integer, providing near-optimal linear bounds.

Contribution

It establishes a nearly tight linear upper bound on the minimal complete graph size for divisible subdivisions, improving previous superexponential bounds.

Findings

Established a linear upper bound of n+8n+14q for f(H,q)

Proved the bound is optimal up to a constant factor

Enhanced understanding of graph minors and subdivisions with divisibility constraints

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.