Rooted $C_5$-Minors

TL;DR

The paper addresses rooted $C_k$-minors in highly connected graphs and proves that every $10$-connected graph is $C_5$-minor-linked, meaning for any five distinct vertices $x_1,\dots,x_5$ there exists a $C_5$-minor rooted on these vertices. It employs the Thomas–Wollan framework of $5$-massed and rigid-separation structures to reduce global connectivity constraints to a small dense subgraph scenario, then derives a contradiction by showing such a subgraph yields the desired $C_5$-minor. Key contributions include tightening the connectivity bound for $C_5$-minor linkage from previously known values and advancing rooted-minor structure theory via a two-stage argument (dense-subgraph extraction followed by minor construction). The results have implications for fine-grained linkage properties and Hadwiger-type minor theory in highly connected graphs.

Abstract

Let $G$ be a graph and $x_1, x_2, \ldots, x_k$ be distinct vertices of $G$. We say $(G,x_1x_2\ldots x_k)$ has a $C_k$-minor or $G$ has a $C_k$-minor rooted at $x_1x_2\ldots x_k$, if there exist pairwise disjoint sets $X_1, X_2, \ldots, X_k\subseteq V(G)$, such that for all $i\in [k]$, $G[X_i]$ is connected, $x_i\in X_i$, and $G$ has an edge between $X_i$ and $X_{i+1}$, where $X_{k+1}=X_k$. When $k=3$ it is easy to determine when $(G,x_1x_2x_3)$ contains a $C_3$-minor. For $k=4$, Robertson, Seymour and Thomas gave a characterization of $(G,x_1x_2x_3x_4)$ with no $C_4$-minor, which, in particular, implies that such $G$ has connectivity at most 5. In this paper, we apply a method of Thomas and Wollan to prove a result, which implies that if $G$ is $10$-connected then, for all distinct vertices $x_1,x_2,x_3,x_4,x_5$ of $G$, $(G,x_1x_2x_3x_4x_5)$ has a $C_5$-minor.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• proof