# zbMATH — the first resource for mathematics

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:
 [1] Bondy, J. A.; Murty, U. S.R., Graph theory, (2008), Springer New York · Zbl 1134.05001 [2] Dirac, G. A., A property of 4-chromatics graphs and some remarks on critical graphs, J. Lond. Math. Soc., 27, 85-92, (1952) · Zbl 0046.41001 [3] Jaeger, F., On circular flows in graphs, (Finite and Infinite Sets (Eger, 1981), Colloquia Mathematica Societatis Janos Bolyai, vol. 37, (1984), North Holland), 391-402 · Zbl 0567.05049 [4] Jaeger, F., Nowhere-zero flow problems, (Beineke, L.; Wilson, R., Selected Topics in Graph Theory, Vol. 3, (1988), Academic Press London, New York), 91-95 [5] Jaeger, F.; Linial, N.; Payan, C.; Tarsi, M., Group connectivity of graphs—a non-homogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B, 56, 165-182, (1992) · Zbl 0824.05043 [6] Kloks, T., Treewidth, computations and approximations, (1994), Springer-Verlag New York · Zbl 0825.68144 [7] Kochol, M., An equivalent version of the 3-flow conjecture, J. Combin. Theory Ser. B, 83, 258-261, (2001) · Zbl 1029.05088 [8] Lai, H.-J., Group connectivity of 3-edge-connected chordal graphs, Graphs Combin., 16, 165-176, (2000) · Zbl 0966.05041 [9] Lai, H.-J., Mod $$(2 p + 1)$$-orientations and $$K_{1, 2 p + 1}$$-decompositions, SIAM J. Discrete Math., 21, 844-850, (2007) · Zbl 1151.05040 [10] Lai, H.-J.; Shao, Y. H.; Wu, H.; Zhou, J., On $$\operatorname{mod}(2 p + 1)$$-orientations of graphs, J. Combin. Theory Ser. B, 99, 399-406, (2009) [11] Lovász, L. M.; Thomassen, C.; Wu, Y.-Z.; Zhang, C.-Q., Nowhere-zero 3-flows and modulo $$k$$-orientations, J. Combin. Theory Ser. B, (2013) · Zbl 1301.05154 [12] Thomassen, C., The weak 3-flow conjecture and the weak circular flow conjecture, J. Combin. Theory Ser. B, 102, 521-529, (2012) · Zbl 1239.05083 [13] Tutte, W. T., A contribution to the theory of chromatical polynomials, Canad. J. Math., 6, 80-91, (1954) · Zbl 0055.17101 [14] Y. Wu, Integer flows and modulo orientations, Ph.D. Dissertation, West Virginia University, 2012.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.