Pan-H-Linked Graphs

LetH be a multigraph, possibly containing loops. AnH-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n0(H) isolated vertices and n1(H) vertices of degree one. In [10] 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 ≥ 3 with δ(G) ≥ n2 , then G is hamiltonian. Bondy [1] extended Dirac’s theorem by showing that if G satisfied the condition δ(G) ≥ n2 then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from [10] in a similar manner. An H-subdivision H in 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.