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Combined degree and connectivity conditions for H-linked graphs
For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for
every injective map {\tau}: V (H) \rightarrow V (G), we can find internally
disjoint paths in G, such that every edge from uv in H corresponds to a {\tau}
(u) - {\tau} (v) path. To guarantee that a G is H-linked, you need a minimum
degree larger than |G|/2. This situation changes, if you know that G has a
certain connectivity k. Depending on k, even a minimum degree independent of
|G| may suffice. Let {\delta}(k, H, N) be the minimum number, such that every
k-connected graph G with |G| = N and {\delta}(G) \geq {\delta}(k, H, N) is
H-linked. We study bounds for this quantity. In particular, we find bounds for
all multigraphs H with at most three edges, which are optimal up to small
additive or multiplicative constants