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). Z(2p+1) 1 satisfying Sigma(v is an element of V(G))b(v) equivalent to 0 (mod 2p+ 1), G always has an orientation D such that the net out-degreeat every vertex v is congruent to b(v) mod 2p+ 1, then G is strongly Z(2p+1)-connected. The graph G' obtained from G by contracting all nontrivial subgraphs that are strongly Z(2s+1)-connected is called the Z(2s+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 Z(3)-connected graphs, we prove that for a simple graph G on n vertices, and for any integers s > 0 and real numbers alpha, beta with 0 < alpha < 1, if for any nonadjacent vertices u, v is an element of V (G), d(G)(u)+ d(G)(v) = an+ beta, then there exists a finite family F(alpha, s) of nonstrongly Z(2s+1)-connected graphs such that either G is strongly Z(2s+1)-connected or the Z(2s+1)-reduction of G is in F(alpha, s).