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Bisections of Graphs Without Short Cycles

  • GENGHUA FAN (a1), JIANFENG HOU (a1) and XINGXING YU (a2)

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.

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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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References

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Bisections of Graphs Without Short Cycles

  • GENGHUA FAN (a1), JIANFENG HOU (a1) and XINGXING YU (a2)

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