AI 中文总结
本文为树积猜想的所有d≥2情形提供反例，通过构造具有度d多项式增长的三次扩张器细分图，推翻了该猜想，仅留d=1为开放情形。
AI 中文摘要
Distel、Gollin、Harvey、Hendrey、Hickingbotham、Mohar和Wood（2023）猜想，度为d多项式增长的图可嵌入到d棵线性增长的树与一个常规模完全图的强积中。Illingworth、Norin和Steiner（2026）近期推翻了该猜想的d=4情形。本文为所有整数d≥2的情形提供了猜想的反例，仅留d=1为唯一开放情形。这些反例是经适当细分的三次扩张器，核心贡献是对每个实数d>1，构造了具有度为d多项式增长且平衡分离器大小为Ω(n^(1-1/d)log n)的三次扩张器细分图，其中n表示顶点数。
英文摘要
Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2023) conjectured that graphs of degree-$d$ polynomial growth can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size complete graph. Very recently, the case $d = 4$ of the conjecture was disproved by Illingworth, Norin and Steiner (2026). In this paper, we provide counterexamples to the conjecture for every integer $d \geq 2$, thus leaving $d=1$ as the only open case. Our counterexamples are appropriately subdivided cubic expanders. Our main contribution is to construct, for every real number $d>1$, subdivisions of cubic expanders with degree-$d$ polynomial growth and whose balanced separators have size $Ω(n^{1-1/d}\log n)$, where $n$ denotes the number of vertices.
Comments11 pages
