Connectivity for Kite-Linked Graphs

TL;DR

This paper proves that every 7-connected graph contains a subdivision of a kite graph for any injection of kite vertices, advancing understanding of connectivity and linkage properties in graph theory.

Contribution

It establishes that the minimum connectivity for guaranteeing a kite subdivision in any graph is 7, improving upon the previous bound of 8.

Findings

Every 7-connected graph is kite-linked.

The exact connectivity threshold for kite linkage is determined as 7.

This result narrows the gap in understanding linkage for graphs with four vertices.

Abstract

For a given graph $H$, a graph $G$ is $H$-linked if, for every injection $\varphi: V(H) \to V(G)$, the graph $G$ contains a subdivision of $H$ with $\varphi(v)$ corresponding to $v$, for each $v\in V(H)$. Let $f(H)$ be the minimum integer $k$ such that every $k$-connected graph is $H$-linked. Among graphs $H$ with at least four vertices, the exact value $f(H)$ is only know when $H$ is a path with four vertices or a cycle with four vertices. A kite is graph obtained from $K_4$ by deleting two adjacent edges, i.e., a triangle together with a pendant edge. Recently, Liu, Rolek and Yu proved that every $8$-connected graph is kite-linked. The exact value of $f(H)$ when $H$ is the kite remains open. In this paper, we settle this problem by showing that every 7-connected graph is kite-linked.