An Asymptotically Tight $t\log t$ Bound for $k$-Connected Subgraphs in Dense $K_t$-Minor-Free Graphs

Abstract

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 $C\geq 1$ such that, for all integers $t\geq 3$ and $k\geq t$, every $K_t$-minor-free graph $G$ with $d(G)\geq Ck$ contains a nonempty $k$-connected subgraph $H$ satisfying $v(H)\leq C^2t\log t$. Thus the structural bound improves from $O(t\log^3 t)$ to $O(t\log t)$. We also give a probabilistic construction showing that the $t\log t$ bound on $v(H)$ is best possible up to a constant factor. Consequently, the graphs occurring in the reduction have order $O(t\log^2 t)$ rather than $O(t\log^4 t)$.