当前位置: X-MOL 学术Comb. Probab. Comput. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Minimizing the number of 5-cycles in graphs with given edge-density
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2019-10-09 , DOI: 10.1017/s0963548319000257
Patrick Bennett , Andrzej Dudek , Bernard Lidický , Oleg Pikhurko

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least $$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$ copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

中文翻译:

在给定边密度的图中最小化 5 循环的数量

受 Razborov 关于图中三角形最小密度工作的启发,我们研究了 5 循环的最小密度C5. 我们证明了每个顺序图n和大小$ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $,其中 k ≥ 3 是一个整数,至少包含$$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \超过 {5{k^4}}}){n^5} + o({n^5})$$的副本C5. 这个界限是最优的,因为匹配的上限由平衡的完全ķ- 分图。证明基于标志代数框架。我们还提供了一个稳定的结果。验证我们的证明不需要 SDP 求解器。
更新日期:2019-10-09
down
wechat
bug