无$H$图中PDKEEPAAA的新结果
New results for MaxCut in $H$-free graphs
摘要：
最大割问题要求图$G$中最大割的大小${\rm mc}(G)$。众所周知，对任何$m$-边图$G$，都有${\rm mc}(G)\ge m/2$，其差${\rm mc}(G)-m/2$称为$G$的盈余.对$H$-free图的剩余的研究是由Erdens和Lovász在70年代发起的，他们特别询问了无三角形图会发生什么。阿隆解决了这个问题，他证明了在无三角形的情况下，盈余是$Ω(m^{4/5})$，并找到了与这个界限相匹配的结构。我们证明了这方面的几个新结果。首先，我们证明了对每个固定的奇$r\ge 3$，任何具有$m$边的$C_r$-free图都有剩余$Ω_r\big(m^{\frac{r+1}{r+2}}\big)$.这是严格的，正如Alon和Kahale构建的伪随机无$C_r$图所示。它改进了几位研究者以前的结果，并补充了Alon，Krivelevich和Sudakov的结果，当$r$是偶数时，这是相同的界限。其次，推广了Alon的结果，允许图中有三角形，并证明了如果三角形的个数比密度相同的随机图中的三角形的个数少一点，则图有大的剩余。对于正则图，我们的盈余界限是尖锐的。第三，证明了具有少量$K_r$副本和平均度$d$的$n$-顶点图有剩余$Ω_r(d^{r-1}/n^{r-3})$，当$d$接近$n$时，如果存在一个有约束的稠密伪随机$K_r$-free图，则剩余$Ω_r(d^{r-1}/n^{r-3})$是紧的.这个结果被用来改善最好的已知下界（作为$m$的函数）的盈余$K_r$-自由图。我们的证明结合了半定规划，概率推理，以及组合和光谱参数的技术。
英文摘要：
The MaxCut problem asks for the size ${\rm mc}(G)$ of a largest cut in a graph $G$. It is well known that ${\rm mc}(G)\ge m/2$ for any $m$-edge graph $G$, and the difference ${\rm mc}(G)-m/2$ is called the surplus of $G$. The study of the surplus of $H$-free graphs was initiated by Erdős and Lovász in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is $Ω(m^{4/5})$, and found constructions matching this bound. We prove several new results in this area. Firstly, we show that for every fixed odd $r\ge 3$, any $C_r$-free graph with $m$ edges has surplus $Ω_r\big(m^{\frac{r+1}{r+2}}\big)$. This is tight, as is shown by a construction of pseudorandom $C_r$-free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when $r$ is even. Secondly, generalizing the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs our bounds on the surplus are sharp. Thirdly, we prove that an $n$-vertex graph with few copies of $K_r$ and average degree $d$ has surplus $Ω_r(d^{r-1}/n^{r-3})$, which is tight when $d$ is close to $n$ provided that a conjectured dense pseudorandom $K_r$-free graph exists. This result is used to improve the best known lower bound (as a function of $m$) on the surplus of $K_r$-free graphs. Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.
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