无三角形图的稀疏近似色剖面

中文摘要 AI 辅助

我们证明了Brandt和Thomassé的四色定理的一个稀疏版本，回答了Allen、Böttcher、Kohayakawa和Roberts提出的一个问题。对于每个固定的$0<γ≤1/10$和每个$p=p(n)∈(0,1]$，渐近几乎必然地，每个生成的无三角形$H⊆G(n,p)$且满足$δ(H)≥(1/3+γ)pn$，可以通过删除至多$\min\{C_γn/p,(1/8+γ)pn^2\}$条边而成为四部图。事实上，删除至多$C_γn/p$条边得到的图允许同态到Andrásfai或Vega图，且证书复杂度至多为$1/(3γ)$。结合随机爆炸图的匹配下界，这一结构结果统一地确定了在$p$下，通过$O(n/p)$次边删除实现$q$-部性的最小度阈值：$q=2$时为$2/5$，$q=3$时为$10/29$，每个固定$q≥4$时为$1/3$。对于每个固定$q≥2$和$\log n/n\ll p\ll n^{-1/2}$，渐近几乎必然地，$G(n,p)$包含一个生成的无三角形子图，其最小度为$(1-o(1))pn$，需要$(1/(2q)+o(1))pn^2$次边删除才能成为$q$-部图，这表明即使在这种更强的度条件下，系数$1/(2q)$也无法改进。

英文摘要

We prove a sparse version of the four-colour theorem of Brandt and Thomassé, answering a question of Allen, Böttcher, Kohayakawa and Roberts. For every fixed $0<γ\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $δ(H)\ge(1/3+γ)pn$ can be made four-partite by deleting at most $\min\{C_γn/p,(1/8+γ)pn^2\}$ edges. In fact, deleting at most $C_γn/p$ edges yields a graph that admits a homomorphism to an Andrásfai or Vega graph with certificate complexity at most $1/(3γ)$. Together with matching lower bounds from random blow-ups, this structural result determines, uniformly in $p$, the minimum-degree thresholds for $q$-partiteness with $O(n/p)$ edge deletions: $2/5$ for $q=2$, $10/29$ for $q=3$, and $1/3$ for every fixed $q\ge4$. For every fixed $q\ge2$ and $\log n/n\ll p\ll n^{-1/2}$, asymptotically almost surely $G(n,p)$ contains a spanning triangle-free subgraph with minimum degree $(1-o(1))pn$ that requires $(1/(2q)+o(1))pn^2$ edge deletions to become $q$-partite, showing that the coefficient $1/(2q)$ cannot be improved even under this stronger degree condition.

发表机构

• Fuzhou University（福州大学）
• East China Normal University（华东师范大学）

机构由 AI 辅助整理，请以论文原文为准。

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18 pages