大奇数围长图的不对称同态阈值

中文摘要 AI 辅助

我们确定了从奇数围长至少为7的图到无三角形图的不对称同态阈值，证明$\delta_{\mathrm{hom}}(\{C_3,C_5\},\{C_3\})=\frac{1}{9}$。等价地，对于每个$\varepsilon>0$，每个具有奇数围长至少为7且最小度至少为$(1/9+\varepsilon)n$的$n$顶点图，都允许一个到大小由$\varepsilon$的函数所界定的无三角形图的同态；而存在奇数围长至少为7且最小度至少为$(1/9-\varepsilon)n$的图，对于这些图，不存在这样的有界大小的无三角形同态像。更一般地，对于每个$t\geq 3$，我们证明$\delta_{\mathrm{hom}}(\{C_3,C_5\},\{K_t\})=\frac{1}{3t}$。特别地，对于每个真单调图类$\mathcal C$，奇数围长至少为7的图允许到$\mathcal C$中有界大小图的同态的阈值是正的。我们进一步将这一现象推广到任意奇数围长：对于每个$k\geq 2$和每个奇数围长至少为$2k-1$的图类的真单调子类$\mathcal C$，奇数围长至少为$2k+3$的图允许到$\mathcal C$中有界大小图的同态的阈值是正的。这些结果否证了Gishboliner、Hurley和Wigderson的猜想，并与相应的零色数阈值结果形成鲜明对比。我们的下界构造基于高维Borsuk图，而匹配的上界使用正则性论证来恢复这些构造背后的结构。

英文摘要

We determine the asymmetric homomorphism threshold from graphs of odd girth at least $7$ to triangle-free graphs, showing that $δ_{\mathrm{hom}}(\{C_3,C_5\},\{C_3\})=\frac{1}{9}$. Equivalently, for every $\varepsilon>0$, every $n$-vertex graph of odd girth at least $7$ and minimum degree at least $(1/9+\varepsilon)n$ admits a homomorphism to a triangle-free graph of size bounded by a function of $\varepsilon$, while there exist graphs of odd girth at least $7$ and minimum degree at least $(1/9-\varepsilon)n$ for which no such bounded-size triangle-free homomorphic image exists. More generally, for every $t\geq 3$, we prove $δ_{\mathrm{hom}}(\{C_3,C_5\},\{K_t\})=\frac{1}{3t}$. In particular, for every proper monotone class $\mathcal C$ of graphs, the threshold for graphs of odd girth at least $7$ to admit a homomorphism to a bounded-size graph in $\mathcal C$ is positive. We further extend this phenomenon to arbitrary odd girth: for every $k\geq 2$ and every proper monotone subclass $\mathcal C$ of the class of graphs of odd girth at least $2k-1$, the threshold for graphs of odd girth at least $2k+3$ to admit a homomorphism to a bounded-size graph in $\mathcal C$ is positive. These results disprove conjectures of Gishboliner, Hurley and Wigderson and exhibit a sharp contrast with the corresponding zero chromatic-threshold results. Our lower-bound constructions are based on high-dimensional Borsuk graphs, while the matching upper bounds use regularity arguments to recover the structure underlying these constructions.

发表机构

• Univ. Bordeaux, CNRS, Bordeaux INP（波尔多大学）
• ETH Zürich（苏黎世联邦理工学院）
• ENS de Lyon, Université Claude Bernard Lyon 1（里昂高等师范学院，里昂第一大学）
• Institute of Science and Technology Austria（奥地利科学技术研究所）

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