Pan- H-Linked Graphs.

Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrarymultigraph of order k, sizem, n₀(H) isolated vertices and n₁(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165-182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H and δ(G) ≥ n+m-k+n₁(H)+2n₀(H)/2 , then G contains a spanning H-subdivision with the same ground set as H. As a corollary to this result, the authors were able to obtain Dirac's famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3with δ(G) ≥ n/2 , then G is hamiltonian. Bondy (J.Comb.Theory Ser.B11:80-84, 1971) extended Dirac's theorem by showing that if G satisfied the condition δ(G) ≥ n/2 then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165-182, 2007) in a similar manner. An H-subdivisionHin G is 1-extendible if there exists an H-subdivision H* with the same ground set as H and ∣H*∣ = ∣H∣ + 1. If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that δ(G) ≥ n+m-k+n₁(H)+2n₀(H)/2 , then G is pan-H-linked. This result is sharp. [ABSTRACT FROM AUTHOR]