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Bisections of Graphs Without Short Cycles

Published online by Cambridge University Press:  28 September 2017

GENGHUA FAN
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian 350116, China (e-mail: fan@fzu.edu.cn)
JIANFENG HOU*
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian 350116, China (e-mail: fan@fzu.edu.cn)
XINGXING YU
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA (e-mail: yu@math.gatech.edu)

Abstract

Bollobás and Scott (Random Struct. Alg.21 (2002) 414–430) asked for conditions that guarantee a bisection of a graph with m edges in which each class has at most (1/4+o(1))m edges. We demonstrate that cycles of length 4 play an important role for this question. Let G be a graph with m edges, minimum degree δ, and containing no cycle of length 4. We show that if (i) G is 2-connected, or (ii) δ ⩾ 3, or (iii) δ ⩾ 2 and the girth of G is at least 5, then G admits a bisection in which each class has at most (1/4+o(1))m edges. We show that each of these conditions are best possible. On the other hand, a construction by Alon, Bollobás, Krivelevich and Sudakov shows that for infinitely many m there exists a graph with m edges and girth at least 5 for which any bisection has at least (1/4−o(1))m edges in one of the two classes.

Type
Paper
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

Partially supported by NSFC grant 11331003.

Corresponding author. Partially supported by NSFC grant 11671087.

§

Partially supported by NSF grants DMS-1265564 and DMS-1600738, and by the Hundred Talents Program of Fujian Province.

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