If a graph G contains two edge-disjoint spanning trees T1, T2 with E(T1) ∪ E(T2) = E(G), we say that G is a double tree and (T1, T2) is a spanning tree factorization of G. We show that every double tree has a spanning tree factorization (T1, T2) such that |dT 1 (v) − dT 2 (v)| ≤ 5 for all v ∈ V (G). This resolves a special case of a conjecture of Kriesell.
We show that the problem of deciding whether a given graph G has a well-balanced orientation $$\vec {G}$$
G
→
such that $$d_{\vec {G}}^+(v)\le \ell (v)$$
d
G
→
+
(
v
)
≤
ℓ
(
v
)
for all $$v \in V(G)$$
v
∈
V
(
G
)
for a given function $$\ell :V(G)\rightarrow \mathbb {Z}_{\ge 0}$$
ℓ
:
V
(
G
)
→
Z
≥
0
is NP-complete. We also prove a similar result for best-balanced orientations. This improves a result of Bernáth, Iwata, Király, Király and Szigeti and answers a question of Frank.
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