7-Connected graphs are 4-ordered

Abstract

A graph G is k-ordered if for any distinct vertices v 1 , v 2 , . . . , v k ∈ V (G), it has a cycle through v 1 , v 2 , . . . , 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 ≤ f (4) ≤ 40. We prove that in fact f (4) = 7.

Fat-triangle linkage and kite-linked graphs

“…X. Yu [25] charaterized all obstructions to P 4 -linked graphs, and showed that 7 ≤ f (P 4 ) ≤ 8, and Ellingham, Plummer, and G. Yu [3] showed that f (P 4 ) = 7. Very recently, McCarty, Wang, and X. Yu [16] showed that f (C 4 ) = 7, that is, 7-connected graphs are 4-ordered, confirming a conjecture of Faudree [4]. Beyond this, very little is known about f (H) for other multigraphs H.…”

“…Lemma 3.1 (McCarty, Wang, and Yu [16]). Let {R 1 , R 2 } be a special (u 1 , u 2 , C, A)-separating pair in a graph G, and let x ∈ int(R 1 ).…”

“…In the study of H-linkage, it is important to find suitable intermediate substructures to work with. For example, X. Yu [25] used "ladders" to study P 4 -linkage, and McCarty, Wang, and X. Yu [16] used "skeletons" to study C 4 -linkage. A fat-triangle is one such substructure that can be useful in the study of some H-linkage problems, in particular when H has few vertices or has a K 3 subgraph.…”

“…Consequently, f (P + 4 ) ∈ {7, 8}. Note that the proof of f (C 4 ) = 7 in [16] relies on the result f (P 4 ) = 7 from [3], whose proof was quite involved. Our proof for kite-linkage is self-contained and follows the ideas of the proof of f (C 4 ) = 7 in [16].…”

“…Note that the proof of f (C 4 ) = 7 in [16] relies on the result f (P 4 ) = 7 from [3], whose proof was quite involved. Our proof for kite-linkage is self-contained and follows the ideas of the proof of f (C 4 ) = 7 in [16]. By using the fat-triangle linkage, we first find a structure called a flower (see Figure 1), and show that every 7-connected graph with a certain specified flower is kite-linked.…”