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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.

##### MSC:
 05C40 Connectivity
Full Text:
##### References:
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