3 research outputs found
Minimum degree conditions for H-linked graphs
AbstractFor a fixed multigraph H with vertices w1,β¦,wm, a graph G is H-linked if for every choice of vertices v1,β¦,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 dβ² such that every n-vertex graph with minimum degree at least dβ² 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 -linked digraphs
Given any digraph , let be the family of all directed
paths in , and let be a digraph with the arc set . The digraph is called arbitrary Hamiltonian -linked if for any
injective mapping and any integer set
with for each , there exists a mapping such that for
every arc , is a directed path from to of length
, and different arcs are mapped into internally vertex-disjoint directed
paths in , and . In this paper, we prove
that for any digraph with arcs and , every digraph of
sufficiently large order with minimum in- and out-degree at least
is arbitrary Hamiltonian -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