On strongly Z2s+1Z2s+1-connected graphs

An orientation of a graph GG is a mod(2s+1)mod(2s+1)-orientation if under this orientation, the net out-degree at every vertex is congruent to zero mod(2s+1)mod(2s+1). If for any function b:V(G)→Z2s+1b:V(G)→Z2s+1 satisfying ∑v∈V(G)b(v)≡0(mod2s+1), GG always has an orientation DD such that the net out-degree at every vertex vv is congruent to b(v)mod(2s+1)b(v)mod(2s+1), then GG is strongly Z2s+1Z2s+1-connected. In this paper, we prove that a connected graph has a mod(2s+1)mod(2s+1)-orientation if and only if it is a contraction of a (2s+1)(2s+1)-regular bipartite graph. We also proved that every (4s−1)(4s−1)-edge-connected series–parallel graph is strongly Z2s+1Z2s+1-connected, and every simple 4p4p-connected chordal graph is strongly Z2s+1Z2s+1-connected.