Abstract
A balanced bipartition of a graph G is a partition of V(G) into two subsets V 1 and V 2 that differ in cardinality by at most 1. A minimum balanced bipartition of G is a balanced bipartition V 1, V 2 of G minimizing e(V 1,V 2), where e(V 1,V 2) is the number of edges joining V 1 and V 2 and is usually referred to as the size of the bipartition. In this paper, we show that every 2-connected graph G admits a balanced bipartition V 1,V 2 such that the subgraphs of G induced by V 1 and by V 2 are both connected. This yields a good upper bound to the size of minimum balanced bipartition of sparse graphs. We also present two upper bounds to the size of minimum balanced bipartitions of triangle-free graphs which sharpen the corresponding bounds of Fan et al. (Discrete Math. 312:1077–1083, 2012).
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References
Bollobás B, Scott AD (2002) Problems and results on judicious partitions. Random Struct Algorithms 21:414–430
Bondy JA, Murty USR (2008) Graph theory. Springer, New York
Fan G, Xu B, Yu X, Zhou C (2012) Upper bounds on minimum balanced bipartitions. Discrete Math 312:1077–1083
Garey MR, Johnson DS, Stockmeyer LJ (1976) Some simplified NP-completed graph problems. Theor Comput Sci 1:237–267
Karpinski M (2002) On approximability of minimum bisection problem. In: Electronic colloquium on computational complexity, Report 46
Lee C, Loh PS, Sudakov B (2011) Bisections of graphs. arXiv:1109.3180
Tutte WT (1947) The factorization of linear graphs. J Lond Math Soc 22:107–111
Xu B, Yan J, Yu X (2010) A note on balanced bipartitions. Discrete Math 310:2613–2617
Acknowledgements
The authors are grateful sincerely to the anonymous referees for their valuable comments and suggestions, which lead to a great improvement of the manuscript.
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M. Liu was partially supported by NSFC project 11071088, the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM10039), and the Project of Graduate Education Innovation of Jiangsu Province (No. CXZZ12-0378).
B. Xu was partially supported by NSFC projects 10931003 and 11171160 and by the Doctoral Fund of Ministry of Education of China.
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Li, H., Liang, Y., Liu, M. et al. On minimum balanced bipartitions of triangle-free graphs. J Comb Optim 27, 557–566 (2014). https://doi.org/10.1007/s10878-012-9539-y
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DOI: https://doi.org/10.1007/s10878-012-9539-y