Fat-triangle linkage and kite-linked graphs

For a multigraph $H$, a graph $G$ is $H$-linked if every injective mapping $\phi: V(H)\to V(G)$ can be extended to an $H$-subdivision in $G$. We study the minimum connectivity required for a graph to be $H$-linked. 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 $8$.