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On large semi-linked graphs. (English) Zbl 1302.05036
Summary: Let \(H\) be a multigraph, possibly with loops, and consider a set \(S \subseteq V(H)\). A (simple) graph \(G\) is \((H, S)\)-semi-linked if, for every injective map \(f : S \to V(G)\), there exists an injective map \(g : V(H) \setminus S \to V(G) \setminus f(S)\) and a set of \(| E(H) |\) internally disjoint paths in \(G\) connecting pairs of vertices of \(f(S) \cup g(V(H) \setminus S)\) for every edge between the corresponding vertices of \(H\). This new concept of \((H, S)\)-semi-linkedness is a generalization of \(H\)-linkedness. We establish a sharp minimum degree condition for a sufficiently large graph \(G\) to be \((H, S)\)-semi-linked.
MSC:
05C07 Vertex degrees
05C35 Extremal problems in graph theory
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