Abstract
Let S = (a 1, …, a m ; b 1, …, b n ), where a 1, …, a m and b 1, …, b n are two nonincreasing sequences of nonnegative integers. The pair S = (a 1, …, a m ; b 1, …, b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y, E) such that a 1, …, a m and b 1, …, b n are the degrees of the vertices in X and Y, respectively. Let Z 3 be the cyclic group of order 3. Define σ(Z 3, m, n) to be the minimum integer k such that every bigraphic pair S = (a 1, …, a m ; b 1, …, b n ) with a m , b n ≥ 2 and σ(S) = a 1 + ⋯ + a m ≥ k has a Z 3-connected realization. For n = m, Yin [Discrete Math., 339, 2018—2026 (2016)] recently determined the values of σ(Z 3, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z 3, m, n) for m ≥ n ≥ 4.
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Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, American Elsevier, New York, 1976
Chen, J. J., Eschen, E., Lai, H. J.: Group connectivity of certain graphs. Ars Combin., 89, 141–158 (2008)
Gale, D.: A theorem on flows in networks. Pac. J. Math., 7, 1073–1082 (1957)
Hakimi, S. L.: On the realizability of a set of integers as degrees of vertices of a graph. J. SIAM Appl. Math., 10, 496–506 (1962)
Havel, V.: A remark on the existence of finite graphs (Czech.). Časopis Pěst. Mat., 80, 477–480 (1955)
Jaeger, F.: Nowhere-zero flow problems, In: Topics in Graph Theory, 3 (L. W. Beineke and R. J. Wilson, editors), Academic Press, London, 1988, 70–95
Jaeger, F., Linial, N., Payan, C., et al.: Group connectivity of graphs—A nonhomogeneous analogue of nowhere zero flow properties. J. Combin. Theory Ser. B, 56, 165–182 (1992)
Kleitman, D. J., Wang, D. L.: Algorithm for constructing graphs and digraphs with given valences and factors. Discrete Math., 6, 79–88 (1973)
Lai, H. J.: Group connectivity of 3-edge-connected chordal graphs. Graphs Combin., 16, 165–176 (2000)
Lai, H. J., Li, X. W., Shao, Y. H., et al.: Group connectivity and group colorings of graphs—A survey. Acta Math. Sin., Engl. Ser., 27, 405–434 (2011)
Luo, R., Xu, R., Yu, G. X.: An extremal problem on group connectivity of graphs. European J. Combin., 33, 1078–1085 (2012)
Luo, R., Xu, R., Zang, W. A., et al.: Realizing degree sequences with graphs having nowhere-zero 3-flows. SIAM J. Discrete Math., 22, 500–519 (2008)
Luo, R., Zang, W. A, Zhang, C. Q.: Nowhere-zero 4-flows, simultaneous edge-colorings, and critical partial Latin squares. Combinatorica, 24, 641–657 (2004)
Ryser, H. J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9, 371–377 (1957)
Yin, J. H.: An extremal problem on bigraphic pairs with an A-connected realization. Discrete Math., 339, 2018–2026 (2016)
Yin, J. H., Guo, G. D.: The smallest degree sum that yields graphic sequences with a Z 3-connected realization. European J. Combin., 34, 806–811 (2013)
Yin, J. H., Luo, R., Guo, G. D.: Graphic sequences with an A-connected realization. Graphs Combin., 30, 1615–1620 (2014)
Zhang, Y., Yin, J. H.: Group connectivity for K −k m, n . Util. Math., 92, 139–147 (2013)
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The authors would like to thank the referees for their helpful suggestions and comments.
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Yin, J.H., Dai, X.Y. Solution to an extremal problem on bigraphic pairs with a Z 3-connected realization. Acta. Math. Sin.-English Ser. 33, 1131–1153 (2017). https://doi.org/10.1007/s10114-017-6380-3
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DOI: https://doi.org/10.1007/s10114-017-6380-3