On Minimum Degree Implying That a Graph is H‐Linked

Given a fixed multigraph $H$, possibly containing loops, with $V(H) = \{h_1,\ldots,h_m\}$, we say that a graph $G$ is $H$-linked if for every choice of $m$ vertices $v_1,\ldots,v_m$ in $G$, there exists a subdivision of $H$ in $G$ such that $v_i$ is the branch vertex representing $h_i$ (for all $i$). This generalizes the concept of $k$-linked graphs (as well as a number of other well-known path or cycle properties). In this paper we determine a sharp lower bound on $\delta(G)$ (which depends upon $H$) such that each graph $G$ on at least $10(|V(H)|+|E(H)|)$ vertices satisfying this bound is $H$-linked.