Let G be a connected graph with XβV(G) and with the spanning
forest F. Let Ξ»β[0,1] be a real number and let Ξ·:Xβ(Ξ»,β) be a real function. In this paper, we show that if for all
SβX, Ο(GβS)β€βvβSβ(Ξ·(v)β2)+2βΞ»(eGβ(S)+1), then G has a spanning tree T
containing F such that for each vertex vβX, dTβ(v)β€βΞ·(v)βΞ»β+max{0,dFβ(v)β1}, where Ο(GβS)
denotes the number of components of GβS and eGβ(S) denotes the
number of edges of G with both ends in S. This is an improvement of several
results and the condition is best possible. Next, we also investigate an
extension for this result and deduce that every k-edge-connected graph G
has a spanning subgraph H containing m edge-disjoint spanning trees such
that for each vertex v, dHβ(v)β€βkmβ(dGβ(v)β2m)β+2m, where kβ₯2m; also if G contains k
edge-disjoint spanning trees, then H can be found such that for each vertex
v, dHβ(v)β€βkmβ(dGβ(v)βm)β+m, where kβ₯m.
Finally, we show that strongly 2-tough graphs, including (3+1/2)-tough
graphs of order at least three, have spanning Eulerian subgraphs whose degrees
lie in the set {2,4}. In addition, we show that every 1-tough graph has
spanning closed walk meeting each vertex at most 2 times and prove a
long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed
walk; connected factor; toughness; total exces