On judicious bipartitions of graphs
نویسندگان
چکیده
For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G) = V1 ∪ V2 such that G has at most s edges with both incident vertices in Vi. Alon proved that the limsup of f(m)− (m/2 + √ m/8) tends to infinity as m tends to infinity, establishing a conjecture of Erdős. Bollobás and Scott proposed the following judicious version of Erdős’ conjecture: the limsup of m/4 + √ m/32− g(m) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon’s result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to k-partitions for odd integers k. ∗School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China. †School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USA. Partially supported by NSF grants DMS-1265564 and AST-1247545 and NSA grant H98230-13-1-0255.
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عنوان ژورنال:
- Combinatorica
دوره 36 شماره
صفحات -
تاریخ انتشار 2016