AI 中文总结
本文研究图的$\lambda$-可选性的阈值，提出以最小$q$-码度替代最小度，推导了阈值$f(\lambda)$的上下界，并对划分为$q$个相等部分的情况渐近确定了阈值，推广了Alon等人的相关结论。
AI 中文摘要
设$\lambda=\{k_1,\ldots,k_q\}$为一个划分，记$|\lambda|=k_1+\cdots+k_q$。图$G$的一个$|\lambda|$-列表分配$L$若满足：其颜色集可划分为$q$个互不相交的集合$X_1,\ldots,X_q$，使得对每个顶点$v$及每个$i\in[q]$，均有$|L(v)\cap X_i|=k_i$，则称该分配为一个$\lambda$-分配。这一概念由Zhu在《J. Combin. Theory Ser. B, 2020》中提出，它将普通着色与列表着色纳入同一框架。Alon在《Random Structures Algorithms, 2000》中证明，每个最小度为$d$的图的可选数至少为$(1/2-o(1))\log_2 d$；Saxton和Thomason在《Invent. Math., 2015》中利用超图容器法将常数$1/2$替换为精确常数$1$。自然可问，对每个固定划分$\lambda$是否存在类似现象？仅靠最小度不够：平衡完全二分图的最小度可任意大，但始终是$\{1,1\}$-可选的。本文证明合适的替代量是最小$q$-码度，对$|V(G)|\geq q$，其定义为$\delta_q(G)=\min\{|N_G(S)|:\\,S\subseteq V(G),\\,|S|=q\}$。更确切地说，对每个划分$\lambda$，存在整数$d$使得每个满足$\delta_q(G)\geq d$的图$G$都不是$\lambda$-可选的，记最小的此类$d$为$f(\lambda)$。对每个固定$q$，当$|\lambda|\to\infty$时，本文证明$f(\lambda)\leq2^{(2q+o(1))|\lambda|}$，同时对所有$\lambda$有$f(\lambda)\geq(q+1)^{-1}(1+1/q)^{|\lambda|}$。对划分为$q$个相等部分的$\{k,\ldots,k\}$，本文渐近确定了阈值：当$k\to\infty$时，$f(\{k,\ldots,k\})=\rho_q^{-(1+o(1))k}$，其中$\rho_q$是满足$x=(1-x)^q$的唯一$x\in(0,1)$。当$q=1$时，本文结果意味着$\operatorname{ch}(G)\geq(1-o(1))\log_2\delta(G)$。
英文摘要
Let $λ=\{k_1,\ldots,k_q\}$ be a partition, and let $|λ|=k_1+\cdots+k_q$. A $|λ|$-list assignment $L$ of a graph $G$ is a $λ$-assignment if its color set can be partitioned into $q$ disjoint sets $X_1,\ldots,X_q$ such that $|L(v)\cap X_i|=k_i$ for every vertex $v$ and every $i\in[q]$. This notion, introduced by Zhu [J. Combin. Theory Ser. B, 2020], puts ordinary coloring and list coloring in the same framework. A theorem of Alon [Random Structures Algorithms, 2000] states that every graph with minimum degree $d$ has choice number at least $(1/2-o(1))\log_2d$. Saxton and Thomason [Invent. Math., 2015] later used the hypergraph container method to replace $1/2$ by the sharp constant $1$. It is natural to ask whether a similar phenomenon holds for every fixed partition $λ$. Minimum degree alone is not sufficient: balanced complete bipartite graphs have arbitrarily large minimum degree but are always $\{1,1\}$-choosable. We show that the appropriate replacement is the minimum $q$-codegree, defined for $|V(G)|\geq q$ by $δ_q(G)=\min\{|N_G(S)|:S\subseteq V(G),\,|S|=q\}$. More precisely, for every partition $λ$ there exists an integer $d$ such that every graph $G$ with $δ_q(G)\geq d$ is not $λ$-choosable. Let $f(λ)$ be the least such $d$. For every fixed $q$, we prove $f(λ)\leq2^{(2q+o(1))|λ|}$ as $|λ|\to\infty$, while $f(λ)\geq(q+1)^{-1}(1+1/q)^{|λ|}$ for every $λ$. For the partition $\{k,\ldots,k\}$ with $q$ equal parts, we determine the threshold asymptotically: $f(\{k,\ldots,k\})=ρ_q^{-(1+o(1))k}$ as $k\to\infty$, where $ρ_q$ is the unique $x\in(0,1)$ satisfying $x=(1-x)^q$. When $q=1$, our result implies $\operatorname{ch}(G)\geq(1-o(1))\log_2δ(G)$.
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