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On strongly \(\mathbb{Z}_{2s + 1}\)-connected graphs. (English) Zbl 1297.05132
Summary: An orientation of a graph \(G\) is a \(\bmod (2s + 1)\)-orientation if under this orientation, the net out-degree at every vertex is congruent to zero \(\bmod (2s+1)\). If for any function \(b : V(G) \to \mathbb{Z}_{2 s + 1}\) satisfying \(\sum_{v \in V(G)} b(v) \equiv 0 \pmod {(2s+1)}\), \(G\) always has an orientation \(D\) such that the net out-degree at every vertex \(v\) is congruent to \(b(v) \bmod (2s+1)\), then \(G\) is strongly \(\mathbb{Z}_{2 s + 1}\)-connected. In this paper, we prove that a connected graph has a \(\bmod (2s+1)\)-orientation if and only if it is a contraction of a \((2 s + 1)\)-regular bipartite graph. We also proved that every \((4 s - 1)\)-edge-connected series-parallel graph is strongly \(\mathbb{Z}_{2 s + 1}\)-connected, and every simple \(4 p\)-connected chordal graph is strongly \(\mathbb{Z}_{2 s + 1}\)-connected.

05C40 Connectivity
Full Text: DOI
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