Fat-triangle linkage and kite-linked graphs

Abstract

For a multigraph H, a graph G is H-linked if every injective mapping φ : V (H) → V (G) can be extended to an H-subdivision in G. We study the minimum connectivity required for a graph to be Hlinked. A k-fat-triangle F k is a multigraph with three vertices and a total of k edges. We determine a sharp connectivity requirement for a graph to be F k -linked. In particular, any k-connected graph is F k -linked when F k is connected. A kite is the graph obtained from K 4 by removing two edges at a vertex. As a nontrivial application of F k -linkage, we then prove that every 8-connected graph is kite-linked, which shows that the required connectivity for a graph to be kite-linked is 7 or 8.

Connectivity for Kite-Linked Graphs

“…In [12], McCarty, Wang and X. Yu show f (C 4 ) = 7, which was originally conjectured by Faudree [4]. Very recently, Liu, Rolek and G. Yu [10] show that every 8-connected graph is kite-linked, where kite is a graph obtained from K 4 by deleting two adjacent edges. So 7 ≤ f (H) ≤ 8 if H is the kite.…”

“…In this section, we are going to prove Theorem 1.3. Before proceeding to the proof, we need some results from [6,14] and [10].…”

“…The following is a useful structure defined by Liu, Rolek and G. Yu in [10] to find a kite-subdivision.…”

“…(2) each path P i , internally disjoint from the three cycles, joins the vertex x i to a vertex v i ∈ C 3 for i ∈ [3] such that v 1 , v 2 , v 3 , x 4 are four different vertices appearing on C 3 in order. Lemma 2.2 (Liu, Rolek and Yu, [10]). If G is a 7-connected graph containing a (x 1 , x 2 , x 3 , x 4 )-flower for some vertices x 1 , x 2 , x 3 , x 4 ∈ V (G), then G contains a kite-subdivision K rooted at x 1 , x 2 , x 3 , x 4 such that x 4 has degree 1 in K and x 2 has degree 3 in K.…”

“…So 7 ≤ f (H) ≤ 8 if H is the kite. The proofs of these results where H is P 4 , C 4 , or the kite in [3,10,12] are based on fine structures of 3-planar graphs and obstructions developed in [16,20,21,22].…”