Connectivity for Kite-Linked Graphs
Runrun Liu, Martin Rolek, D. Christopher Stephens, Dong Ye, Gexin Yu · SIAM Journal on Discrete Mathematics · 2021
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 connected simple graphs $H$ with at least four vertices, the exact value $f(H)$ is only known when $H$ is a star, or a path with four vertices, or a cycle with four vertices. A kite is the graph obtained from $K_4$ by deleting two adjacent edges, i.e., a triangle together with a pendant edge. 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.