标题： 忠实的泛图对于子式闭包类 显示英文标题

标题： Faithful universal graphs for minor-closed classes

摘要： 由Huynh、Mohar、Šámal、Thomassen和Wood在2021年证明，任何包含每个可数平面图作为子图的可数图都有无限团小结构。我们证明了这个结果的有限定量版本：对于固定的$t$，如果图$G$是$K_t$-小结构自由的，并且包含每个$n$顶点平面图作为子图，那么$G$有$2^{\Omega(n)}$个顶点。 另一方面，我们构造了一个多项式大小的$K_4$-不含子图的图，其中包含每个$n$-顶点树作为诱导子图，并且构造了一个多项式大小的$K_7$-不含子图的图，其中包含每个$n$-顶点$K_4$-不含子图的图作为诱导子图。 这回答了Bergold、Iršič、Lauff、Orthaber、Scheucher和Wesolek最近提出的一些问题。 我们更广泛地研究了各种类别的通用图的阶数（有界度、树深度、路径宽度或树宽的图），如果通用图保留了原始类的一些结构。 显示英文摘要

摘要： It was proved by Huynh, Mohar, \v{S}\'amal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for fixed $t$, if a graph $G$ is $K_t$-minor-free and contains every $n$-vertex planar graph as a subgraph, then $G$ has $2^{\Omega(n)}$ vertices. On the other hand, we construct a polynomial size $K_4$-minor-free graph containing every $n$-vertex tree as an induced subgraph, and a polynomial size $K_7$-minor-free graph containing every $n$-vertex $K_4$-minor-free graph as induced subgraph. This answers several problems raised recently by Bergold, Ir\v{s}i\v{c}, Lauff, Orthaber, Scheucher and Wesolek. We study more generally the order of universal graphs for various classes (of graphs of bounded degree, treedepth, pathwidth, or treewidth), if the universal graphs retain some of the structure of the original class.