3 research outputs found

    Minimum degree conditions for H-linked graphs

    Get PDF
    AbstractFor a fixed multigraph H with vertices w1,…,wm, a graph G is H-linked if for every choice of vertices v1,…,vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and nβ©Ύ7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer dβ€² such that every n-vertex graph with minimum degree at least dβ€² is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H

    A Dirac-type theorem for arbitrary Hamiltonian HH-linked digraphs

    Full text link
    Given any digraph DD, let P(D)\mathcal{P}(D) be the family of all directed paths in DD, and let HH be a digraph with the arc set A(H)={a1,…,ak}A(H)=\{a_1, \ldots, a_k\}. The digraph DD is called arbitrary Hamiltonian HH-linked if for any injective mapping f:V(H)β†’V(D)f: V(H)\rightarrow V(D) and any integer set N={n1,…,nk}\mathcal{N}=\{n_1, \ldots, n_k\} with niβ‰₯4n_i\geq4 for each i∈{1,…,k}i\in\{1, \ldots, k\}, there exists a mapping g:A(H)β†’P(D)g: A(H)\rightarrow \mathcal{P}(D) such that for every arc ai=uva_i=uv, g(ai)g(a_i) is a directed path from f(u)f(u) to f(v)f(v) of length nin_i, and different arcs are mapped into internally vertex-disjoint directed paths in DD, and ⋃i∈[k]V(g(ai))=V(D)\bigcup_{i\in[k]}V(g(a_i))=V(D). In this paper, we prove that for any digraph HH with kk arcs and Ξ΄(H)β‰₯1\delta(H)\geq1, every digraph of sufficiently large order nn with minimum in- and out-degree at least n/2+kn/2+k is arbitrary Hamiltonian HH-linked. Furthermore, we show that the lower bound is best possible. Our main result extends some work of K\"{u}hn and Osthus et al. \cite{20081,20082} and Ferrara, Jacobson and Pfender \cite{Jacobson}. Besides, as a corollary of our main theorem, we solve a conjecture of Wang \cite{Wang} for sufficiently large graphs
    corecore