Proper Minor-Closed Classes in Blowups of Fans

TL;DR

This paper proves that any $K_t$-minor-free graph with n vertices can be embedded into a blowup of a fan with size proportional to _t(\u221a n ^2(n)), extending previous results from planar graphs to more general minor-closed classes.

Contribution

It generalizes the embedding of planar graphs into blowups of fans to all $K_t$-minor-free graphs, establishing a bound with explicit dependence on t.

Findings

Any $K_t$-minor-free graph is contained in a $O_t(\u221a n ^2(n))$-blowup of a fan.

Extends previous planar graph results to broader minor-closed classes.

Provides bounds that depend explicitly on t for these embeddings.

Abstract

Dujmovi\'{c} et al. [arXiv:2407.05936] showed that any $n$-vertex planar graph is contained in a $O(\sqrt{n}\log^2(n))$-blowup of a fan, and asked if the same holds for any $n$-vertex $K_t$-minor-free graph. We answer this in the positive, showing that such a graph is contained in a $O_t(\sqrt{n}\log^2(n))$-blowup of a fan.