Edge-connectivity and the orientation of a graph
Panagiotis Katerinis, N. Tsikopoulos · SUT Journal of Mathematics · 2005
Let G be a k-edge-connected graph and let L denote the subset of all vertices having odd degree in G. For every subset K={u1,u2,…,uk} of L with |K|≤|L|2, and for every function h defined on K having the property that h(ui)∈{ ⌈ dG(ui)2 ⌉,⌊ dG(ui)2 ⌋ } for all ui∈K, there exists an orientation D of G such that dD+(x)=h(x) when x∈K and ⌊ dG(x)2 ⌋≤dD+(x)≤⌈ dG(x)2 ⌉ when x∈V(G)−K.