2016/10/14 by Miaomiao Han, Hong-Jian Lai, Han, Miaomiao +3
Mathematics · #05C21 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C21
paper · pdf · doi:10.48550/arxiv.1610.04581
14 pages, 3 figures
arxiv created 2016/10/14 · arxiv updated 2016/10/17
Given a zero-sum function β: V(G) → ℤ3 with ∑v∈ V(G)β(v)=0, an orientation D of G with d+D(v)-d-D(v)= β(v) in ℤ3 for every vertex v∈ V(G) is called a β-orientation. A graph G is ℤ3-connected if G admits a β- orientation for every zero-sum function β. Jaeger et al. conjectured that every 5-edge-connected graph is ℤ3-connected. A graph is ⟨ℤ3⟩-extendable at vertex v if any pre-orientation at v can be extended to a β-orientation of G for any zero-sum function β. We observe that if every 5-edge-connected essentially 6-edge-connected graph is ⟨ℤ3⟩-extendable at any degree five vertex, then the above mentioned conjecture by Jaeger et al. holds as well. Furthermore, applying the partial flow extension method of Thomassen and of Lovász et al., we prove that every graph with at least 4 edge-disjoint spanning trees is ℤ3-connected. Consequently, every 5-edge-connected essentially 23-edge-connected graph is ⟨ℤ3⟩-extendable at degree five vertex.