On Mod (2s+1)-Orientations of Graphs

An orientation of a graph G is a mod (2p+1)-orientation if, under this orientation, the net out-degree at every vertex is congruent to zero mod 2p+1. If, for any function b : V (G) → Z2p+1 satisfying ∑ v∈V (G) b(v) ≡ 0 (mod 2p + 1), G always has an orientation D such that the net outdegree at every vertex v is congruent to b(v) mod 2p + 1, then G is strongly Z2p+1-connected. The graph G′ obtained from G by contracting all nontrivial subgraphs that are strongly Z2s+1connected is called the Z2s+1-reduction of G. Motivated by a minimum degree condition of Barat and Thomassen [J. Graph Theory, 52 (2006), pp. 135–146], and by the Ore conditions of Fan and Zhou [SIAM J. Discrete Math., 22 (2008), pp. 288–294] and of Luo et al. [European J. Combin., 29 (2008), pp. 1587–1595] on Z3-connected graphs, we prove that for a simple graph G on n vertices, and for any integers s > 0 and real numbers α, β with 0 < α < 1, if for any nonadjacent vertices u, v ∈ V (G), dG(u)+ dG(v) ≥ αn+β, then there exists a finite family F(α, s) of nonstrongly Z2s+1connected graphs such that either G is strongly Z2s+1-connected or the Z2s+1-reduction of G is in F(α, s).