AI 中文总结
本文针对局部稀疏图，证明满足特定边数条件的图中存在大量导出偶环，部分解决了Ding等人提出的相关问题。
AI 中文摘要
对于c>0且t≥1，若图Γ中任意一对顶点子集A,B⊆V(Γ)满足|A|,|B|≥t时，它们之间的边数e(A,B)≤(1−c)|A||B|，则称Γ为(c,t)-稀疏图。本文证明：对任意整数ℓ≥2，存在ε>0、C、C′>0，若n顶点图Γ是(1−ε,t)-稀疏的且边数至少为Ct^(1−1/ℓ)n^(1+1/ℓ)，则Γ包含至少C′n²t^(2ℓ−2)个导出的2ℓ阶环C_{2ℓ}，这部分解决了Ding、Gao、Liu、Luan和Sun提出的问题。
英文摘要
A graph $Γ$ is $(c,t)$-sparse for $c > 0$ and $t \ge 1$ if for every pair of vertex subsets $A, B \subseteq V(Γ)$ with $|A|, |B| \ge t$, the number of edges $e(A,B)$ between them satisfies $ e(A,B) \le (1 - c)|A||B|$. In this paper, we prove that for every integer $\ell\ge2$, there are $\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $Γ$ is $(1-\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\ell}n^{1+1/\ell}$ edges, then $Γ$ contains at least $C'n^2t^{2\ell-2}$ induced copies of $C_{2\ell}$. This partially resolves a problem of Ding, Gao, Liu, Luan, and Sun.
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