Ore-type degree conditions for a graph to be H-linked

Given a fixed multigraph H with V(H) = {h1, . . . ,hm}, we say that a graph G is H-linked if for every choice of m vertices v1, . . . , vm in G, there exists a subdivision of H in G such that for every i, vi is the branch vertex representing hi . This generalizes the notion of k-linked graphs (as well as some other notions). For a family H of graphs, a graph G is H-linked if G is H-linked for every H ∈ H. In this article, we estimate the minimum integer r = r(n, k,d) such that each n-vertex graph with σ2(G) ≥ r is H-linked, where H is the Contract grant sponsor: NSF (to A. V. K.); Contract grant numbers: DMS-0400498, DMS-0650784; Contract grant sponsor: RFBR; Contract grant number: 05-0100816; Contract grant sponsor: NSF (to G. Y.); Contract grant number: DMS0652306. Journal of Graph Theory © 2008 Wiley Periodicals, Inc.