An $O(t\log^2 t)$ Bound for $k$-Connected Subgraphs in Dense $K_t$-Minor-Free Graphs

Delcourt and Postle reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices.

An important theorem in their proof process asserts that every sufficiently dense $K_t$-minor-free graph contains a small, highly connected subgraph.

In this paper, we show that such a subgraph can be chosen to be smaller.

More precisely, there exists an integer constant $C \ge 1$ such that, for all integers $t \ge 3$ and $k \ge t$, every $K_t$-minor-free graph $G$ with $d(G) \ge Ck$ contains a nonempty $k$-connected subgraph $H$ satisfying $v(H) \le C^2 t\log^2 t$.

Thus the structural bound improves from $O(t\log^3 t)$ to $O(t\log^2 t)$, and the graphs occurring in the reduction have order $O(t\log^3 t)$ rather than $O(t\log^4 t)$.