Abstract
Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.
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Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)
Catlin, P.A.: Double cycle covers and the Petersen graph. J. Graph Theory 13, 465–483 (1989)
Catlin, P.A.: The reduction of graph families under contraction. Discrete Math. 160, 67–80 (1996)
Catlin, P.A., Hobbs, A.M., Lai, H.-J.: Graph families operations. Discrete Math. 230, 71–97 (2001)
Fan, G., Lai, H.-J., Xu, R., Zhang, C.-Q., Zhou, C.: Nowhere-zero \(3\)-flows in triangularly connected graphs. J. Comb. Theory Ser. B 98(6), 1325–1336 (2008)
Jaeger, F.: Flows and generalized coloring theorems in graphs. J. Comb. Theory Ser. B 26, 205–216 (1979)
Jaeger, F.: Nowhere-zero flow problems. In: Beineke, L., Wilson, R. (eds.) Selected Topics in Graph Theory 3, pp. 71–95. Wiley, New York (1988)
Jaeger, F., Linial, N., Payan, C., Tarsi, M.: Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties. J. Comb. Theory Ser. B 56, 165–182 (1992)
Lai, H.-J.: Extending a partial nowhere zero 4-flow. J. Graph Theory 30, 277–288 (1999)
Lai, H.-J., Lai, H.Y.: Duality of graph families. Discrete Math. 110, 165–177 (1992)
Lai, H.-J., Liang, Y., Liu, J., Meng, J., Miao, Z., Shao, Y., Zhang, Z.: On strongly \({\mathbb{Z}}_{2s+1}\)-connected graphs. Discrete Appl. Math. 174, 73–80 (2014)
Lovász, L.M., Thomassen, C., Wu, Y.-Z., Zhang, C.-Q.: Nowhere-zero 3-flows and modulo \(k\)-orientations. J. Comb. Theory Ser. B 103, 587–598 (2013)
Seymour, P.D.: Nowhere-zero \(6\)-flows. J. Comb. Theory Ser. B 30, 130–136 (1981)
Steinberg, R., Younger, D.H.: Grötzsch’s theorem for the projective plane. Ars Comb. 28, 15–31 (1989)
Thomassen, C.: The weak 3-flow conjecture and the weak circular flow conjecture. J. Comb. Theory Ser. B 102, 521–529 (2012)
Tutte, W.T.: On the embedding of linear graphs in surfaces. Proc. Lond. Math. Soc. Ser. 2(51), 474–483 (1949)
Tutte, W.T.: A contribution to the theory of chromatical polynomials. Can. J. Math. 6, 80–91 (1954)
Zhang, C.-Q.: Integer Flows and Cycle Covers of Graphs. Marcel Dekker Inc., New York (1997). ISBN 0-8247-9790-6
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Liang, Y., Lai, HJ., Luo, R. et al. Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations. Graphs and Combinatorics 32, 1065–1075 (2016). https://doi.org/10.1007/s00373-015-1636-0
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DOI: https://doi.org/10.1007/s00373-015-1636-0