Faithful universal graphs for minor-closed classes
For graph theorists studying minor-closed classes, this provides tight bounds on the size of universal graphs under structural constraints, resolving open problems.
The paper proves a quantitative version of a result by Huynh et al.: any K_t-minor-free graph containing all n-vertex planar graphs as subgraphs must have 2^{Ω(n)} vertices. Conversely, they construct polynomial-size universal graphs for trees (K_4-minor-free) and K_4-minor-free graphs (K_7-minor-free), answering several open problems.
It was proved by Huynh, Mohar, Šámal, 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 $2^{Ω(n)}$ 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, Iršič, 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.