7-Connected Graphs are 4-Ordered

TL;DR

This paper proves that every 7-connected graph is 4-ordered, establishing the exact value of the connectivity threshold for this property, which was previously only bounded between 7 and 40.

Contribution

The paper determines that the minimum connectivity for 4-ordered graphs is exactly 7, resolving a longstanding open problem.

Findings

Proves that $f(4)=7$ for 4-ordered graphs.

Establishes the exact connectivity threshold for 4-ordered property.

Improves previous bounds from 7 to 40 to an exact value.

Abstract

A graph $G$ is $k$-ordered if for any distinct vertices $v_1, v_2, \ldots, v_k \in V(G)$, it has a cycle through $v_1, v_2, \ldots, v_k$ in order. Let $f(k)$ denote the minimum integer so that every $f(k)$-connected graph is $k$-ordered. The first non-trivial case of determining $f(k)$ is when $k=4$, where the previously best known bounds are $7 \leq f(4) \leq 40$. We prove that in fact $f(4)=7$.