On strongly Z2s+1 Z 2 s + 1 -connected graphs
An orientation of a graph G G is a mod(2s+1) mod ( 2 s + 1 ) -orientation if under this orientation, the net out-degree at every vertex is congruent to zero mod(2s+1) mod ( 2 s + 1 ) . If for any function b:V(G)?Z2s+1 b : V ( G ) ? Z 2 s + 1 satisfying ? v ? V ( G ) b ( v ) ? 0 ( mod 2 s + 1 ) , G G always has an orientation D D such that the net out-degree at every vertex v v is congruent to b(v)mod(2s+1) b ( v ) mod ( 2 s + 1 ) , then G G is strongly Z2s+1 Z 2 s + 1 -connected. In this paper, we prove that a connected graph has a mod(2s+1) mod ( 2 s + 1 ) -orientation if and only if it is a contraction of a (2s+1) ( 2 s + 1 ) -regular bipartite graph. We also proved that every (4s?1) ( 4 s ? 1 ) -edge-connected seriesparallel graph is strongly Z2s+1 Z 2 s + 1 -connected, and every simple 4p 4 p -connected chordal graph is strongly Z2s+1 Z 2 s + 1 -connected.
mod(2s+1) mod ( 2 s + 1 ) -orientations mod(2s+1) mod ( 2 s + 1 ) mod(2s+1) mod ( 2 s + 1 ) mod ( 2 s + 1 ) nowhere zero flows integer flows unitary zm z m -flows zm z m zm z m z m strongly z2s+1 z 2 s + 1 -connectedness z2s+1 z 2 s + 1 z2s+1 z 2 s + 1 z 2 s + 1 group connectivity